Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C2×#575 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C22⋊C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,10,16,11,15,31,43,12,17,32,44,9,27,30,55)(7,18,22,46,21,13,33,36,19,37,48,61,28,34,56,59)(8,14,25,40,24,41,53,65,26,42,54,66,29,39,57,64)(20,45,49,67,50,35,58,60,47,38,62,63,23,51,52,68);(5,7)(6,13)(8,23)(9,28)(10,22)(11,21)(12,19)(14,38)(15,37)(16,36)(17,34)(18,27)(20,24)(25,52)(26,50)(29,47)(30,56)(31,33)(32,48)(35,39)(40,63)(41,51)(42,45)(43,61)(44,59)(46,55)(49,53)(54,58)(57,62)(60,64)(65,68)(66,67);(2,4)(7,19)(13,34)(18,37)(20,47)(21,28)(22,48)(23,50)(33,56)(35,51)(36,59)(38,45)(46,61)(49,62)(52,58)(60,68)(63,67);(5,8)(6,14)(7,20)(9,29)(10,25)(11,24)(12,26)(13,35)(15,41)(16,40)(17,42)(18,45)(19,47)(21,50)(22,49)(23,28)(27,39)(30,57)(31,53)(32,54)(33,58)(34,51)(36,60)(37,38)(43,65)(44,66)(46,67)(48,62)(52,56)(55,64)(59,68)(61,63)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=ci,d^b=de,e^b=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404472);