Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C16 · Schur multiplier (H2) = C29 |
| direct product | C22×#2258 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32 | C22 |
| commutator | C2 | C24×C16 |
| Frattini, cosocle | C16 | C25 |
| socle | C23 | C22×C16 |
| prime radical | C22×#41 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2);(3,4);(5,6,8,13,9,14,21,31,10,15,22,32,24,34,43,53,11,16,23,33,25,35,44,54,26,36,45,55,46,56,61,66)(7,12,18,28,19,29,40,50,20,30,41,51,42,52,60,65,17,27,37,47,38,48,57,62,39,49,58,63,59,64,67,68);(5,7)(6,12)(8,18)(9,19)(10,20)(11,17)(13,28)(14,29)(15,30)(16,27)(21,40)(22,41)(23,37)(24,42)(25,38)(26,39)(31,50)(32,51)(33,47)(34,52)(35,48)(36,49)(43,60)(44,57)(45,58)(46,59)(53,65)(54,62)(55,63)(56,64)(61,67)(66,68);(7,17)(12,27)(18,37)(19,38)(20,39)(28,47)(29,48)(30,49)(40,57)(41,58)(42,59)(50,62)(51,63)(52,64)(60,67)(65,68)320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532377);