Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #33982 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D16 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C22×#357 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 272 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,6,12,8,15,33,14)(3,16,19,57,22,58,21,69)(4,10,25,44,23,42,71,46)(5,27,30,80,32,82,87,81)(7,34,29,13,36,52,86,54)(9,37,39,65,41,64,55,18)(11,47,49,88,51,103,50,31)(17,56,61,108,59,107,63,114)(20,67,66,104,68,105,35,48)(24,73,76,120,70,115,118,121)(26,78,75,45,72,100,117,95)(28,53,83,93,85,106,84,40)(38,89,91,109,90,111,102,60)(43,96,98,119,94,125,99,77)(62,113,112,126,110,128,79,97)(74,101,122,124,116,127,123,92);(1,3)(2,9)(4,17)(5,28)(6,19)(7,35)(8,22)(10,38)(11,48)(12,39)(13,53)(14,55)(15,41)(16,27)(18,31)(20,29)(21,33)(23,59)(24,74)(25,61)(26,79)(30,83)(32,85)(34,40)(36,66)(37,47)(42,90)(43,97)(44,91)(45,101)(46,102)(49,67)(50,105)(51,104)(52,93)(54,106)(56,73)(57,80)(58,82)(60,77)(62,75)(63,71)(64,103)(65,88)(68,86)(69,81)(70,116)(72,112)(76,122)(78,92)(84,87)(89,96)(94,126)(95,127)(98,113)(99,128)(100,124)(107,115)(108,120)(109,119)(110,117)(111,125)(114,121)(118,123);(1,4)(2,10)(3,17)(5,24)(6,25)(7,26)(8,23)(9,38)(11,43)(12,44)(13,45)(14,46)(15,42)(16,56)(18,60)(19,61)(20,62)(21,63)(22,59)(27,73)(28,74)(29,75)(30,76)(31,77)(32,70)(33,71)(34,78)(35,79)(36,72)(37,89)(39,91)(40,92)(41,90)(47,96)(48,97)(49,98)(50,99)(51,94)(52,100)(53,101)(54,95)(55,102)(57,108)(58,107)(64,111)(65,109)(66,112)(67,113)(68,110)(69,114)(80,120)(81,121)(82,115)(83,122)(84,123)(85,116)(86,117)(87,118)(88,119)(93,124)(103,125)(104,126)(105,128)(106,127);(4,23)(10,42)(17,59)(24,70)(25,71)(26,72)(38,90)(43,94)(44,46)(45,95)(56,107)(60,109)(61,63)(62,110)(73,115)(74,116)(75,117)(76,118)(77,119)(78,100)(79,112)(89,111)(91,102)(92,124)(96,125)(97,126)(98,99)(101,127)(108,114)(113,128)(120,121)(122,123)Polycyclic presentation:
2^9|a^2=f,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320305, #72320813, #72320873 |
| maximal normal subgroup of | #72320305, #72320813, #72320873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419891);