Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421767 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15297 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,28)(3,14,17,9,18,34,19,33)(4,10,29,37,23,38,22,41)(5,24,26,11,27,42,30,40)(15,45,47,31,20,43,48,35)(16,49,51,44,50,32,25,39)(21,53,55,36,56,58,57,62)(46,60,54,61,52,59,64,63);(1,3,8,17,7,18,6,19)(2,9,12,34,13,33,28,14)(4,20,29,48,23,15,22,47)(5,25,26,16,27,51,30,50)(10,35,37,45,38,31,41,43)(11,39,42,49,40,44,24,32)(21,54,55,52,56,64,57,46)(36,61,58,59,62,63,53,60);(1,4,7,23)(2,10,13,38)(3,15,18,20)(5,21,27,56)(6,29,8,22)(9,31,33,35)(11,36,40,62)(12,41,28,37)(14,43,34,45)(16,46,50,52)(17,48,19,47)(24,58,42,53)(25,54,51,64)(26,57,30,55)(32,59,49,60)(39,61,44,63);(1,5)(2,11)(3,16)(4,21)(6,30)(7,27)(8,26)(9,32)(10,36)(12,42)(13,40)(14,44)(15,46)(17,51)(18,50)(19,25)(20,52)(22,57)(23,56)(24,28)(29,55)(31,59)(33,49)(34,39)(35,60)(37,58)(38,62)(41,53)(43,63)(45,61)(47,54)(48,64)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,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, 328964);