Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473267 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16857 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
652 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)(3,8)(4,18)(5,23,25,10)(6,27)(7,12)(9,21)(11,13)(14,41)(15,30,45,53)(16,47)(17,32)(19,49)(20,51,24,50)(22,38)(26,37,40,54)(28,35)(29,43)(31,39)(33,48)(34,52,36,55)(42,60,44,62)(46,64,59,58)(56,63,57,61);(1,3,13,39,7,17,6,16)(2,8,11,31,12,32,27,47)(4,19,28,43,22,14,21,48)(5,15,40,59,25,45,26,46)(9,33,18,49,35,29,38,41)(10,30,37,58,23,53,54,64)(20,44,55,61,24,42,52,63)(34,57,50,62,36,56,51,60);(1,4,7,22)(2,9,12,35)(3,14,17,19)(5,24,25,20)(6,28,13,21)(8,29,32,33)(10,36,23,34)(11,38,27,18)(15,44,45,42)(16,48,39,43)(26,52,40,55)(30,57,53,56)(31,49,47,41)(37,50,54,51)(46,61,59,63)(58,60,64,62);(1,5,6,26,7,25,13,40)(2,10,11,37,12,23,27,54)(3,15,16,46,17,45,39,59)(4,20,21,52,22,24,28,55)(8,30,31,58,32,53,47,64)(9,34,18,50,35,36,38,51)(14,42,43,61,19,44,48,63)(29,56,41,60,33,57,49,62)Polycyclic presentation:
2^9|a^2=e,b^2=efgh,c^2=i,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380464);