Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461299 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16386 |
| 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 | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,13,6,12)(3,15,18,33,19,34,30,9)(4,20,24,37,23,10,22,40)(5,25,29,11,28,41,27,39)(16,44,46,35,48,42,47,31)(17,49,51,59,53,32,52,43)(21,38,56,36,55,61,26,54)(45,57,50,58,63,60,64,62);(1,3,6,18,7,19,8,30)(2,9,14,34,13,33,12,15)(4,16,22,46,23,48,24,47)(5,17,27,51,28,53,29,52)(10,35,40,44,20,31,37,42)(11,32,41,59,39,49,25,43)(21,45,26,50,55,63,56,64)(36,60,61,58,54,57,38,62);(1,4)(2,10)(3,16)(5,26)(6,24)(7,23)(8,22)(9,31)(11,38)(12,40)(13,20)(14,37)(15,42)(17,50)(18,47)(19,48)(21,27)(25,36)(28,56)(29,55)(30,46)(32,58)(33,35)(34,44)(39,61)(41,54)(43,57)(45,51)(49,62)(52,63)(53,64)(59,60);(1,5,6,27,7,28,8,29)(2,11,12,25,13,39,14,41)(3,17,18,51,19,53,30,52)(4,21,22,26,23,55,24,56)(9,32,15,43,33,49,34,59)(10,36,37,38,20,54,40,61)(16,45,46,50,48,63,47,64)(31,57,44,58,35,60,42,62)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368495);