Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150144 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2140 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12731 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
616 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 104 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 — 3 generators on 128 points:
(1,2)(3,15)(4,21,23,10)(5,26)(6,33)(7,37)(8,12)(9,17)(11,30)(13,29)(14,35)(16,53,57,99)(18,55)(19,64)(20,65)(22,69,28,86)(24,76,74,96)(25,45,75,68)(27,51)(31,94)(32,79)(34,46)(36,97)(38,54)(39,56,66,60)(40,85)(41,84)(42,61)(43,72,47,88)(44,71,67,87)(48,93)(49,110)(50,91)(52,62)(58,117,116,128)(59,118,78,114)(63,111)(70,113,82,105)(73,125,89,122)(77,107,119,103)(80,90)(81,92)(83,112)(95,102)(98,108)(100,123,121,126)(101,120,106,115)(104,127,109,124);(1,3)(2,9)(4,16)(5,27)(6,34)(7,38)(8,18)(10,39)(11,46)(12,42)(13,51)(14,52)(15,26)(17,30)(19,29)(20,35)(21,66)(22,70)(23,57)(24,77)(25,78)(28,82)(31,95)(32,83)(33,40)(36,98)(37,41)(43,103)(44,105)(45,106)(47,107)(48,108)(49,111)(50,112)(53,69)(54,94)(55,79)(56,72)(58,71)(59,75)(60,88)(61,93)(62,110)(63,91)(64,80)(65,81)(67,113)(68,101)(73,126)(74,119)(76,121)(84,90)(85,92)(86,99)(87,116)(89,123)(96,100)(97,102)(104,117)(109,128)(114,125)(115,127)(118,122)(120,124);(1,4)(2,10)(3,16)(5,28)(6,24)(7,25)(8,23)(9,39)(11,47)(12,21)(13,44)(14,45)(15,53)(17,60)(18,57)(19,58)(20,59)(22,32)(26,69)(27,82)(29,71)(30,88)(31,89)(33,96)(34,77)(35,75)(36,74)(37,68)(38,78)(40,100)(41,101)(42,66)(43,48)(46,107)(49,109)(50,67)(51,105)(52,106)(54,114)(55,99)(56,61)(62,120)(63,116)(64,117)(65,118)(70,83)(72,93)(73,92)(76,97)(79,86)(80,104)(81,122)(84,115)(85,126)(87,91)(90,127)(94,125)(95,123)(98,119)(102,121)(103,108)(110,124)(111,128)(112,113)Polycyclic presentation:
2^9|a^2=f,d^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,e^b=ei,g^b=gh,h^b=hi,d^c=di,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72320367, #72320813, #72320868 |
| maximal normal subgroup of | #72320367, #72320813, #72320868 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59363);