Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152284 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12813 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #509 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
257 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 257 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,16,20,43)(4,22)(5,11)(6,31)(7,36)(8,15)(9,18,41,66)(10,24)(12,34)(13,29)(14,30)(17,59,62,121)(19,68,32,94)(21,42,71,110)(23,75)(25,82)(26,77)(27,76)(28,87)(33,54)(35,74)(37,51)(38,49)(39,101,104,122)(40,106,50,114)(44,78)(45,98)(46,85)(47,81)(48,90)(52,118)(53,80)(55,83)(56,79,102,61)(57,97,109,70)(58,96,100,67)(60,103,64,73)(63,115,123,107)(65,92,88,108)(69,120,95,105)(72,119,124,128)(84,116)(86,117)(89,113)(91,93)(99,112)(111,125,127,126);(1,3,8,20)(2,9)(4,17,26,62)(5,18,30,66)(6,32,35,19)(7,21,38,71)(10,39)(11,16)(12,50)(13,42)(14,43)(15,41)(22,73)(23,60,27,64)(24,79,81,61)(25,63,84,123)(28,88,91,65)(29,67,83,96)(31,92)(33,95,99,69)(34,97,80,70)(36,100)(37,72,86,124)(40,53)(44,102)(45,105)(46,56)(47,104)(48,109)(49,58)(51,115)(52,57)(54,111)(55,110)(59,75)(68,93)(74,108)(76,121)(77,103)(78,101,85,122)(82,128)(87,94)(89,126,98,125)(90,114,118,106)(107,117)(112,127)(113,120)(116,119);(1,4,8,26)(2,10,15,47)(3,17,20,62)(5,27,30,23)(6,33,35,99)(7,37,38,86)(9,39,41,104)(11,46,14,44)(12,51,53,117)(13,54,55,112)(16,56,43,102)(18,64,66,60)(19,69,32,95)(21,72,71,124)(22,74,77,31)(24,80,81,34)(25,83,84,29)(28,89,91,98)(36,45,49,113)(40,107,50,115)(42,111,110,127)(48,82,52,116)(57,119,109,128)(58,120,100,105)(59,94,121,68)(61,97,79,70)(63,96,123,67)(65,125,88,126)(73,108,103,92)(75,87,76,93)(78,90,85,118)(101,114,122,106)Polycyclic presentation:
2^9|a^2=f,b^2=dh,c^2=i,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eg,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59481);