Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16310 |
| 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 | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,42)(4,20,25,37)(5,26,30,11)(6,13,7,12)(9,31,34,18)(10,23,38,24)(16,45,48,32)(17,49,52,44)(21,35,47,43)(22,53,55,39)(27,57,54,36)(28,40,29,41)(33,51,56,59)(46,62,58,60)(50,61,64,63);(1,3,6,18,8,19,7,31)(2,9,12,15,14,34,13,42)(4,21,24,16,25,47,23,48)(5,17,28,51,30,52,29,59)(10,35,20,32,38,43,37,45)(11,33,40,44,26,56,41,49)(22,50,54,46,55,64,27,58)(36,61,53,60,57,63,39,62);(1,4,6,24,8,25,7,23)(2,10,12,20,14,38,13,37)(3,16,18,47,19,48,31,21)(5,27,28,22,30,54,29,55)(9,32,15,43,34,45,42,35)(11,39,40,36,26,53,41,57)(17,50,51,46,52,64,59,58)(33,61,44,60,56,63,49,62);(1,5,6,28,8,30,7,29)(2,11,12,40,14,26,13,41)(3,17,18,51,19,52,31,59)(4,22,23,27,25,55,24,54)(9,33,15,44,34,56,42,49)(10,36,37,39,38,57,20,53)(16,46,21,50,48,58,47,64)(32,60,35,61,45,62,43,63)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365324);