Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #433518 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15483 |
| 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 | |
727 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,14,6,13)(3,15,19,34,18,9,20,41)(4,21,24,10,25,37,30,36)(5,26,31,11,29,39,28,40)(16,43,46,32,22,51,45,42)(17,47,48,38,27,58,49,33)(23,53,54,57,55,35,56,50)(44,60,63,64,52,59,62,61);(1,3,6,20,7,18,8,19)(2,9,12,41,14,15,13,34)(4,22,24,45,25,16,30,46)(5,27,28,48,29,17,31,49)(10,32,21,43,36,42,37,51)(11,38,39,58,40,33,26,47)(23,44,54,63,55,52,56,62)(35,60,57,61,53,59,50,64);(1,4)(2,10)(3,16,18,22)(5,23)(6,30)(7,25)(8,24)(9,32,15,42)(11,35)(12,37)(13,21)(14,36)(17,44,27,52)(19,46,20,45)(26,57)(28,56)(29,55)(31,54)(33,59,38,60)(34,43,41,51)(39,50)(40,53)(47,61,58,64)(48,63,49,62);(1,5,7,29)(2,11,14,40)(3,17,18,27)(4,23,25,55)(6,31,8,28)(9,33,15,38)(10,35,36,53)(12,26,13,39)(16,44,22,52)(19,49,20,48)(21,50,37,57)(24,56,30,54)(32,59,42,60)(34,47,41,58)(43,61,51,64)(45,63,46,62)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340715);