Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434015 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15495 |
| 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 | |
900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,7,13,8,12)(3,15,18,9,20,33,19,34)(4,10,24,37,25,38,23,42)(5,26,28,11,30,40,29,41)(16,35,21,50,47,43,46,31)(17,32,27,44,48,57,49,39)(22,52,56,62,55,58,54,36)(45,60,59,63,53,64,51,61);(1,3,6,18,7,20,8,19)(2,9,12,15,13,34,14,33)(4,21,23,16,25,46,24,47)(5,27,28,48,30,49,29,17)(10,35,37,50,38,43,42,31)(11,39,26,57,41,44,40,32)(22,53,54,59,55,45,56,51)(36,60,52,63,62,64,58,61);(1,4,8,23,7,25,6,24)(2,10,14,37,13,38,12,42)(3,16,18,21,20,47,19,46)(5,22,28,56,30,55,29,54)(9,31,15,43,34,50,33,35)(11,36,26,58,41,62,40,52)(17,45,49,51,48,53,27,59)(32,60,44,63,57,64,39,61);(1,5,7,30)(2,11,13,41)(3,17,20,48)(4,22,25,55)(6,29,8,28)(9,32,34,57)(10,36,38,62)(12,40,14,26)(15,44,33,39)(16,45,47,53)(18,49,19,27)(21,51,46,59)(23,56,24,54)(31,60,50,64)(35,61,43,63)(37,58,42,52)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=ef,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341212);