Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #33982 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D16 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C2×C4×D16 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,6,13,8,11,33,15)(3,16,19,58,22,57,21,69)(4,10,25,43,23,46,71,45)(5,27,30,81,32,80,87,82)(7,34,29,14,36,53,86,48)(9,37,40,65,39,64,55,18)(12,49,51,88,47,103,52,31)(17,56,61,107,59,108,63,114)(20,67,66,104,68,106,35,50)(24,73,76,120,70,115,118,121)(26,78,75,44,72,99,117,101)(28,54,83,93,85,105,84,41)(38,89,90,109,92,111,102,60)(42,94,96,119,98,125,97,77)(62,113,112,126,110,127,79,95)(74,100,122,124,116,128,123,91);(1,3)(2,9)(4,17)(5,28)(6,19)(7,35)(8,22)(10,38)(11,39)(12,50)(13,40)(14,54)(15,55)(16,27)(18,31)(20,29)(21,33)(23,59)(24,74)(25,61)(26,79)(30,83)(32,85)(34,41)(36,66)(37,49)(42,95)(43,90)(44,100)(45,102)(46,92)(47,104)(48,105)(51,67)(52,106)(53,93)(56,73)(57,80)(58,81)(60,77)(62,75)(63,71)(64,103)(65,88)(68,86)(69,82)(70,116)(72,112)(76,122)(78,91)(84,87)(89,94)(96,113)(97,127)(98,126)(99,124)(101,128)(107,120)(108,115)(109,119)(110,117)(111,125)(114,121)(118,123);(1,4,6,25,8,23,33,71)(2,10,13,43,11,46,15,45)(3,17,19,61,22,59,21,63)(5,24,30,76,32,70,87,118)(7,26,29,75,36,72,86,117)(9,38,40,90,39,92,55,102)(12,42,51,96,47,98,52,97)(14,44,53,99,48,101,34,78)(16,56,58,107,57,108,69,114)(18,60,37,89,65,109,64,111)(20,62,66,112,68,110,35,79)(27,73,81,120,80,115,82,121)(28,74,83,122,85,116,84,123)(31,77,49,94,88,119,103,125)(41,91,54,100,93,124,105,128)(50,95,67,113,104,126,106,127);(2,11)(4,23)(9,39)(12,47)(13,15)(14,48)(16,57)(17,59)(24,70)(25,71)(26,72)(27,80)(34,53)(40,55)(41,93)(50,104)(51,52)(54,105)(58,69)(60,109)(61,63)(62,110)(67,106)(74,116)(75,117)(76,118)(77,119)(79,112)(81,82)(89,111)(94,125)(122,123)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,d^a=di,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, #72320842, #72320882, … |
| maximal normal subgroup of | #72320305, #72320842, #72320882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419971);