Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C25 |
| direct product | C2×#13737 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C64 | C22 |
| commutator | C2 | C23×C32 |
| Frattini, cosocle | C32 | C24 |
| socle | C22 | C22×C32 |
| prime radical | C2×#41 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 130 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 130 points:
(1,2);(3,4,6,12,7,13,22,37,8,14,23,38,26,41,57,77,9,15,24,39,27,42,58,78,29,44,60,80,63,83,97,112,10,16,25,40,28,43,59,79,30,45,61,81,64,84,98,113,31,46,62,82,65,85,99,114,66,86,100,115,101,116,122,128)(5,11,18,33,19,34,51,71,20,35,52,72,54,74,93,108,21,36,53,73,55,75,94,109,56,76,95,110,96,111,121,127,17,32,47,67,48,68,87,102,49,69,88,103,90,105,117,123,50,70,89,104,91,106,118,124,92,107,119,125,120,126,129,130);(3,5)(4,11)(6,18)(7,19)(8,20)(9,21)(10,17)(12,33)(13,34)(14,35)(15,36)(16,32)(22,51)(23,52)(24,53)(25,47)(26,54)(27,55)(28,48)(29,56)(30,49)(31,50)(37,71)(38,72)(39,73)(40,67)(41,74)(42,75)(43,68)(44,76)(45,69)(46,70)(57,93)(58,94)(59,87)(60,95)(61,88)(62,89)(63,96)(64,90)(65,91)(66,92)(77,108)(78,109)(79,102)(80,110)(81,103)(82,104)(83,111)(84,105)(85,106)(86,107)(97,121)(98,117)(99,118)(100,119)(101,120)(112,127)(113,123)(114,124)(115,125)(116,126)(122,129)(128,130);(5,17)(11,32)(18,47)(19,48)(20,49)(21,50)(33,67)(34,68)(35,69)(36,70)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(93,117)(94,118)(95,119)(96,120)(108,123)(109,124)(110,125)(111,126)(121,129)(127,130)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420503);