Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,17,12,16,10,33)(7,18,23,43,21,13,22,36)(8,14,28,40,26,39,27,44)(9,29,32,58,30,15,31,42)(19,34,47,37,48,45,24,52)(20,41,50,65,51,35,49,60)(25,38,55,63,56,57,54,66)(46,64,68,62,67,59,53,61);(1,2)(3,4)(5,7)(6,13)(8,24)(9,20)(10,23)(11,22)(12,21)(14,37)(15,41)(16,18)(17,43)(19,27)(25,53)(26,47)(28,48)(29,35)(30,51)(31,50)(32,49)(33,36)(34,40)(38,62)(39,52)(42,60)(44,45)(46,54)(55,67)(56,68)(57,61)(58,65)(59,66)(63,64);(2,4)(5,8)(6,14,16,39)(7,19)(9,25)(10,28)(11,27)(12,26)(13,34,18,45)(15,38,29,57)(17,44,33,40)(20,46)(21,48)(22,47)(23,24)(30,56)(31,55)(32,54)(35,59,41,64)(36,52,43,37)(42,66,58,63)(49,68)(50,53)(51,67)(60,62,65,61);(5,9,12,30)(6,15)(7,20,21,51)(8,25,26,56)(10,31,11,32)(13,35)(14,38)(16,29)(17,42)(18,41)(19,46,48,67)(22,49,23,50)(24,53,47,68)(27,54,28,55)(33,58)(34,59)(36,60)(37,61)(39,57)(40,63)(43,65)(44,66)(45,64)(52,62)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290960);