Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| 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 | |
1588 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)(7,19)(8,25,29,14)(9,30)(10,18)(11,16)(12,17)(13,22)(15,31)(20,39,47,53)(21,48)(23,37)(24,38)(26,35,52,62)(27,56,58,59)(28,41,34,44)(32,42)(33,43)(36,50)(40,57,54,60)(45,49)(46,65,66,61)(51,63)(55,64,67,68);(1,2,4,3)(5,7,10,22,12,23,11,24)(6,13,16,19,17,38,18,37)(8,26,34,20,29,52,28,47)(9,21,31,49,33,50,32,51)(14,39,44,35,25,53,41,62)(15,36,30,45,42,48,43,63)(27,55,60,46,58,67,57,66)(40,64,56,61,54,68,59,65);(5,8)(6,14)(7,20)(9,27)(10,28)(11,34)(12,29)(13,35)(15,40)(16,41)(17,25)(18,44)(19,39)(21,46)(22,26)(23,47)(24,52)(30,59)(31,57)(32,60)(33,58)(36,61)(37,53)(38,62)(42,54)(43,56)(45,64)(48,65)(49,55)(50,66)(51,67)(63,68);(1,3,4,2)(5,9,11,32,12,33,10,31)(6,15,18,43,17,42,16,30)(7,21,24,51,23,50,22,49)(8,27,34,60,29,58,28,57)(13,36,37,63,38,48,19,45)(14,40,44,56,25,54,41,59)(20,46,52,67,47,66,26,55)(35,61,53,68,62,65,39,64)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272264);