Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14745 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
937 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,17,38)(4,18,22,9)(5,23,25,10)(6,26,27,11)(8,28,31,16)(14,39,19,36)(15,42,44,37)(20,33,51,45)(21,34,47,46)(24,35,54,53)(29,41,32,49)(30,43,52,55)(40,59,48,60)(50,62,63,58)(56,64,57,61);(1,3,6,16,7,17,27,28)(2,8,11,13,12,31,26,38)(4,19,47,41,22,14,21,49)(5,15,24,43,25,44,54,55)(9,32,46,36,18,29,34,39)(10,30,35,37,23,52,53,42)(20,48,63,61,51,40,50,64)(33,57,62,59,45,56,58,60);(1,4,6,21,7,22,27,47)(2,9,11,34,12,18,26,46)(3,14,16,41,17,19,28,49)(5,20,24,50,25,51,54,63)(8,29,13,36,31,32,38,39)(10,33,35,58,23,45,53,62)(15,40,43,61,44,48,55,64)(30,56,37,59,52,57,42,60);(1,5,6,24,7,25,27,54)(2,10,11,35,12,23,26,53)(3,15,16,43,17,44,28,55)(4,20,21,50,22,51,47,63)(8,30,13,37,31,52,38,42)(9,33,34,58,18,45,46,62)(14,40,41,61,19,48,49,64)(29,56,36,59,32,57,39,60)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304310);