Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17290 |
| 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 | |
593 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)(3,9,16,34)(4,19,24,38)(5,25,28,42)(6,13)(7,29)(8,12)(10,22,37,23)(11,27,41,30)(14,44)(15,39)(17,33,18,43)(20,51)(21,36)(26,32)(31,47)(35,46)(40,56)(45,58,48,63)(49,59)(50,60)(52,54)(53,62,64,61)(55,57);(1,3)(2,9,12,34)(4,20)(5,26,28,50)(6,18)(7,17)(8,16)(10,35,37,44)(11,39)(13,43,29,33)(14,22)(15,27,49,30)(19,51,38,31)(21,45,56,48)(23,46)(24,47)(25,32)(36,58)(40,63)(41,59)(42,60)(52,61)(53,54,64,55)(57,62);(1,4,6,22,8,24,7,23)(2,10,29,38,12,37,13,19)(3,14,18,47,16,46,17,20)(5,21,27,54,28,56,30,55)(9,31,33,44,34,51,43,35)(11,40,42,52,41,36,25,57)(15,48,26,53,49,45,50,64)(32,58,39,62,60,63,59,61);(1,5)(2,11,12,41)(3,15,16,49)(4,21)(6,27)(7,30)(8,28)(9,32)(10,36,37,40)(13,25,29,42)(14,45,46,48)(17,50,18,26)(19,52,38,57)(20,53,47,64)(22,54)(23,55)(24,56)(31,58)(33,39)(34,60)(35,61)(43,59)(44,62)(51,63)Polycyclic presentation:
2^9|a^2=efg,b^2=gh,c^2=ef,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387059);