Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C8 · Schur multiplier (H2) = C214 |
| direct product | C2×#62624 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C24 |
| commutator | C2 | C25×C8 |
| Frattini, cosocle | C8 | C26 |
| socle | C22 | C24×C8 |
| prime radical | C2×#585 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
7880 conjugacy classes of subgroups; only the 7730 normal subgroups are shown.
This group has 7880 conjugacy classes of subgroups — too many to draw. Only its 7730 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 66 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 66 points:
(1,2);(3,4,7,12,8,13,22,32,9,14,23,33,24,34,44,54)(5,10,17,27,18,28,40,50,16,26,38,48,39,49,58,63)(6,11,20,30,21,31,43,53,19,29,41,51,42,52,59,64)(15,25,36,46,37,47,57,62,35,45,55,60,56,61,65,66);(3,5)(4,10)(6,15)(7,17)(8,18)(9,16)(11,25)(12,27)(13,28)(14,26)(19,35)(20,36)(21,37)(22,40)(23,38)(24,39)(29,45)(30,46)(31,47)(32,50)(33,48)(34,49)(41,55)(42,56)(43,57)(44,58)(51,60)(52,61)(53,62)(54,63)(59,65)(64,66);(3,6)(4,11)(5,15)(7,20)(8,21)(9,19)(10,25)(12,30)(13,31)(14,29)(16,35)(17,36)(18,37)(22,43)(23,41)(24,42)(26,45)(27,46)(28,47)(32,53)(33,51)(34,52)(38,55)(39,56)(40,57)(44,59)(48,60)(49,61)(50,62)(54,64)(58,65)(63,66);(6,19)(11,29)(15,35)(20,41)(21,42)(25,45)(30,51)(31,52)(36,55)(37,56)(43,59)(46,60)(47,61)(53,64)(57,65)(62,66);(5,16)(10,26)(15,35)(17,38)(18,39)(25,45)(27,48)(28,49)(36,55)(37,56)(40,58)(46,60)(47,61)(50,63)(57,65)(62,66)272 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493043);