Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222272 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,3)(5,6,11,18,12,16,10,17)(7,13,21,37,22,36,32,43)(8,23,27,40,28,14,26,41)(9,29,30,44,31,15,33,42)(19,45,24,53,47,34,52,38)(20,48,50,60,51,35,49,61)(25,39,56,63,57,54,58,64)(46,59,66,68,67,65,55,62);(1,2)(3,4)(5,7)(6,13)(8,24)(9,20)(10,32)(11,21)(12,22)(14,38)(15,35)(16,36)(17,43)(18,37)(19,26)(23,53)(25,55)(27,47)(28,52)(29,48)(30,50)(31,51)(33,49)(34,40)(39,62)(41,45)(42,61)(44,60)(46,56)(54,68)(57,66)(58,67)(59,63)(64,65);(5,8,12,28)(6,14,16,23)(7,19,22,47)(9,25,31,57)(10,27,11,26)(13,34,36,45)(15,39,29,54)(17,41,18,40)(20,46,51,67)(21,52,32,24)(30,58,33,56)(35,59,48,65)(37,53,43,38)(42,64,44,63)(49,66,50,55)(60,68,61,62);(1,3,4,2)(5,9,12,31)(6,15,16,29)(7,20,22,51)(8,25,28,57)(10,30,11,33)(13,35,36,48)(14,39,23,54)(17,42,18,44)(19,46,47,67)(21,49,32,50)(24,55,52,66)(26,56,27,58)(34,59,45,65)(37,60,43,61)(38,62,53,68)(40,63,41,64)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290685);