Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17093 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
985 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)(3,15,20,34,19,41,18,9)(4,21,23,36,25,10,24,42)(5,26,31,11,30,39,29,40)(16,45,47,53,48,43,52,32)(17,33,27,37,51,61,50,44)(22,55,28,54,56,38,57,35)(46,59,49,62,58,63,64,60);(1,3,6,18,7,19,8,20)(2,9,12,15,14,34,13,41)(4,16,23,47,25,48,24,52)(5,27,29,17,30,50,31,51)(10,32,36,43,21,53,42,45)(11,37,39,61,40,44,26,33)(22,49,28,58,56,64,57,46)(35,62,38,59,54,60,55,63);(1,4,8,24,7,25,6,23)(2,10,13,42,14,21,12,36)(3,16,20,52,19,48,18,47)(5,28,31,22,30,57,29,56)(9,32,41,45,34,53,15,43)(11,38,26,35,40,55,39,54)(17,49,27,46,51,64,50,58)(33,60,44,59,61,62,37,63);(1,5)(2,11)(3,17)(4,22)(6,31)(7,30)(8,29)(9,33)(10,35)(12,26)(13,39)(14,40)(15,44)(16,46)(18,27)(19,51)(20,50)(21,54)(23,57)(24,28)(25,56)(32,59)(34,61)(36,55)(37,41)(38,42)(43,62)(45,60)(47,64)(48,58)(49,52)(53,63)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383802);