Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #13742 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13743 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D32 |
| commutator, Frattini, cosocle | C32 | C24 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 130 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 130 points:
(1,2);(3,4)(5,17)(6,23)(7,30)(8,35)(9,39)(10,12)(11,18)(13,26)(14,25)(15,32)(16,37)(19,63)(20,67)(21,70)(22,56)(24,51)(27,83)(28,87)(29,55)(31,48)(33,95)(34,89)(36,57)(38,97)(40,58)(41,62)(42,75)(43,74)(44,100)(45,59)(46,64)(47,68)(49,109)(50,111)(52,115)(53,60)(54,61)(65,126)(66,125)(69,127)(71,78)(72,81)(73,85)(76,116)(77,106)(79,91)(80,90)(82,99)(84,113)(86,120)(88,114)(92,110)(93,112)(94,104)(96,108)(98,119)(101,129)(102,128)(103,130)(105,123)(107,124)(117,122)(118,121);(3,5,10,22)(4,11,12,41)(6,24,29,77)(7,31,34,94)(8,36,38,98)(9,40,37,21)(13,48,45,104)(14,51,46,106)(15,53,47,107)(16,54,39,44)(17,55,56,23)(18,59,62,26)(19,64,66,25)(20,68,69,32)(27,84,86,130)(28,88,85,76)(30,42,89,101)(33,96,79,93)(35,43,97,102)(49,110,105,126)(50,112,99,108)(52,116,90,114)(57,120,119,83)(58,73,70,87)(60,123,124,109)(61,82,100,111)(63,117,125,71)(65,80,92,115)(67,118,127,72)(74,122,128,78)(75,121,129,81)(91,113,95,103);(4,12)(11,41)(13,45)(14,46)(15,47)(16,39)(17,56)(23,55)(30,89)(35,97)(42,101)(43,102)(44,54)(48,104)(49,105)(50,99)(51,106)(52,90)(53,107)(57,119)(58,70)(63,125)(67,127)(71,117)(72,118)(73,87)(83,120)(91,95)(103,113)(108,112)(110,126)(114,116)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420510);