Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153609 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13661 |
| commutator, Frattini, cosocle | C2×C32 | C23 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #504 ◁ #2258 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
178 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 42 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)(3,18)(4,24)(5,16)(6,35)(7,40)(8,13)(9,17)(10,19)(11,20)(12,28)(14,26)(15,37)(21,71)(22,73)(23,57)(25,60)(27,43)(29,93)(30,75)(31,78)(32,51)(33,99)(34,64)(36,52)(38,102)(39,103)(41,66)(42,59)(44,87)(45,81)(46,80)(47,69)(48,79)(49,100)(50,74)(53,91)(54,116)(55,88)(56,92)(58,85)(61,84)(62,86)(63,68)(65,67)(70,101)(72,122)(76,83)(77,90)(82,114)(89,112)(94,119)(95,123)(96,110)(97,117)(98,118)(104,113)(105,111)(106,126)(107,128)(108,125)(109,124)(115,127)(120,121);(1,3)(2,10)(4,25)(5,20)(6,36)(7,41)(8,32)(9,23)(11,43)(12,52)(13,44)(14,60)(15,63)(16,65)(17,47)(18,24)(19,28)(21,26)(22,37)(27,79)(29,94)(30,96)(31,82)(33,100)(34,74)(35,45)(38,105)(39,104)(40,46)(42,98)(48,61)(49,55)(50,112)(51,75)(53,111)(54,117)(56,113)(57,78)(58,110)(59,109)(62,114)(64,120)(66,93)(67,88)(68,116)(69,92)(70,84)(71,76)(72,86)(73,77)(80,83)(81,90)(85,97)(87,91)(89,125)(95,128)(99,119)(101,106)(102,107)(103,108)(115,122)(118,123)(121,126)(124,127);(1,3)(2,11)(4,19)(5,32)(6,21)(7,22)(8,20)(9,23)(10,43)(12,48)(13,57)(14,49)(15,50)(16,18)(17,51)(24,65)(25,28)(26,36)(27,87)(29,68)(30,67)(31,69)(33,97)(34,98)(35,101)(37,41)(38,70)(39,72)(40,66)(42,74)(44,78)(45,106)(46,93)(47,75)(52,61)(53,114)(54,108)(55,60)(56,110)(58,113)(59,118)(62,111)(63,112)(64,73)(71,99)(76,119)(77,120)(79,91)(80,90)(81,83)(82,92)(84,105)(85,100)(86,104)(88,96)(89,124)(94,116)(95,121)(102,122)(103,117)(107,115)(109,123)(125,127)(126,128)Polycyclic presentation:
2^9|d^2=fg,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fg,g^b=gh,h^b=hi,e^c=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72321604 |
| maximal normal subgroup of | #72321604 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60805);