Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2234 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13381 |
| prime radical | #33617 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 260 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)(3,15)(4,22)(5,28)(6,33)(7,37)(8,13)(9,17)(10,25)(11,32)(12,35)(14,31)(16,57)(18,64)(19,43)(20,42)(21,44)(23,76)(24,79)(26,75)(27,84)(29,74)(30,90)(34,53)(36,73)(38,50)(39,87)(40,101)(41,105)(45,88)(46,97)(47,82)(48,80)(49,113)(51,81)(52,85)(54,59)(55,67)(56,66)(58,118)(60,120)(61,72)(62,94)(63,99)(65,117)(68,122)(69,123)(70,116)(71,109)(77,102)(78,106)(83,114)(86,92)(89,115)(91,111)(93,95)(96,103)(98,112)(100,127)(104,119)(107,125)(108,126)(110,121)(124,128);(1,3)(2,9)(4,23)(5,20)(6,18)(7,19)(8,21)(10,40)(11,44)(12,41)(13,42)(14,43)(15,28)(16,26)(17,32)(22,72)(24,58)(25,77)(27,78)(29,61)(30,68)(31,66)(33,94)(34,96)(35,67)(36,69)(37,99)(38,100)(39,71)(45,54)(46,103)(47,104)(48,102)(49,108)(50,106)(51,107)(52,109)(53,110)(55,86)(56,87)(57,74)(59,82)(60,83)(62,93)(63,85)(64,90)(65,98)(70,89)(73,122)(75,118)(76,79)(80,119)(81,126)(84,116)(88,101)(91,121)(92,125)(95,123)(97,124)(105,113)(111,117)(112,128)(114,127)(115,120);(1,4,8,26)(2,10,13,47)(3,16)(5,29,32,24)(6,34,36,98)(7,38,39,89)(9,40)(11,48,28,45)(12,50,51,115)(14,53,52,112)(15,54)(17,61)(18,65)(19,70)(20,58)(21,23)(22,73,75,33)(25,81,82,35)(27,85,83,31)(30,91,93,97)(37,46,87,111)(41,106)(42,104)(43,110)(44,102)(49,114,86,84)(55,116)(56,117)(57,64)(59,67)(60,66)(62,121)(63,78)(68,124)(69,96)(71,100)(72,94)(74,95,79,90)(76,123)(77,126)(80,92,88,113)(99,103)(101,105)(107,120)(108,127)(109,128)(118,122)(119,125)Polycyclic presentation:
2^9|c^2=dh,d^2=i,e^2=gh,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,d^c=di,e^c=eg,f^c=fi,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60454);