Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153259 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13381 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
257 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 257 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,8,14)(3,15,21,43)(4,22,28,78)(5,29,32,11)(6,33,36,75)(7,37,39,87)(9,20,44,17)(10,45,50,25)(12,52,54,35)(13,55,57,31)(16,60,64,119)(18,67,72,124)(19,42,74,108)(23,79,61,118)(24,77,26,76)(27,84,85,112)(30,88,91,93)(34,56,96,110)(38,53,83,116)(40,99,102,80)(41,103,106,122)(46,109,100,62)(47,81,48,82)(49,89,111,95)(51,90,86,114)(58,113,105,70)(59,66,97,69)(63,121,98,104)(65,92,71,123)(68,117,126,101)(73,115,120,128)(94,107,125,127);(1,3)(2,9)(4,23)(5,20)(6,18)(7,19)(8,21)(10,46)(11,43)(12,41)(13,42)(14,44)(15,29)(16,26)(17,32)(22,60)(24,64)(25,80)(27,73)(28,61)(30,71)(31,69)(33,92)(34,94)(35,70)(36,72)(37,97)(38,98)(39,74)(40,48)(45,99)(47,102)(49,107)(50,100)(51,105)(52,113)(53,115)(54,106)(55,66)(56,117)(57,108)(58,86)(59,87)(62,82)(63,83)(65,91)(67,88)(68,89)(75,123)(76,118)(77,79)(78,119)(81,109)(84,121)(85,120)(90,103)(93,124)(95,126)(96,125)(101,110)(104,112)(111,127)(114,122)(116,128);(1,4,8,28)(2,10,14,50)(3,16)(5,26,32,24)(6,34,36,96)(7,38,39,83)(9,40)(11,48,29,47)(12,53,54,116)(13,56,57,110)(15,46)(17,23)(18,68)(19,73)(20,61)(21,64)(22,75,78,33)(25,52,45,35)(27,55,85,31)(30,89,91,95)(37,49,87,111)(41,104)(42,107)(43,100)(44,102)(51,112,86,84)(58,115)(59,117)(60,67)(62,70)(63,69)(65,94)(66,98)(71,125)(72,126)(74,120)(76,93,77,88)(79,92)(80,122)(81,90,82,114)(97,101)(99,103)(105,128)(106,121)(108,127)(109,113)(118,123)(119,124)Polycyclic presentation:
2^9|a^2=i,c^2=dh,d^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=di,f^b=fi,d^c=di,e^c=eg,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60456);