Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
775 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)(5,6,11,18,12,16,10,17)(7,19,23,38,24,13,22,39)(8,25,29,14,30,43,28,42)(9,15,34,45,33,47,35,46)(20,40,51,48,52,62,26,36)(21,49,55,63,54,37,31,44)(27,56,58,41,32,60,59,66)(50,64,68,65,53,67,57,61);(5,7,12,24)(6,13,16,19)(8,26,30,51)(9,31,33,55)(10,23,11,22)(14,40,42,62)(15,44,47,63)(17,39,18,38)(20,29,52,28)(21,34,54,35)(25,48,43,36)(27,53,32,50)(37,46,49,45)(41,65,66,61)(56,67,60,64)(57,58,68,59);(5,8,12,30)(6,14,16,42)(7,20,24,52)(9,32,33,27)(10,29,11,28)(13,36,19,48)(15,41,47,66)(17,43,18,25)(21,53,54,50)(22,51,23,26)(31,57,55,68)(34,58,35,59)(37,61,49,65)(38,40,39,62)(44,67,63,64)(45,56,46,60);(5,9,10,35,12,33,11,34)(6,15,17,46,16,47,18,45)(7,21,22,31,24,54,23,55)(8,27,28,59,30,32,29,58)(13,37,38,63,19,49,39,44)(14,41,25,56,42,66,43,60)(20,50,26,57,52,53,51,68)(36,61,62,67,48,65,40,64)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298797);