Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419192 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15268 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,14)(3,15,30,34,19,33,18,9)(4,10,24,36,23,37,22,41)(5,25,29,11,28,39,27,40)(16,42,46,31,20,50,45,58)(17,47,49,43,48,32,26,38)(21,52,56,60,55,61,54,35)(44,63,53,62,51,57,64,59);(1,3,6,18,7,19,8,30)(2,9,12,15,13,34,14,33)(4,20,22,46,23,16,24,45)(5,26,27,48,28,49,29,17)(10,31,36,50,37,58,41,42)(11,38,39,47,40,43,25,32)(21,53,54,44,55,64,56,51)(35,59,52,63,60,62,61,57);(1,4)(2,10)(3,16)(5,21)(6,24)(7,23)(8,22)(9,31)(11,35)(12,41)(13,37)(14,36)(15,42)(17,44)(18,46)(19,20)(25,52)(26,53)(27,56)(28,55)(29,54)(30,45)(32,57)(33,50)(34,58)(38,62)(39,61)(40,60)(43,59)(47,63)(48,51)(49,64);(1,5,7,28)(2,11,13,40)(3,17,19,48)(4,21,23,55)(6,29,8,27)(9,32,34,47)(10,35,37,60)(12,25,14,39)(15,43,33,38)(16,44,20,51)(18,49,30,26)(22,56,24,54)(31,57,58,63)(36,61,41,52)(42,59,50,62)(45,53,46,64)Polycyclic presentation:
2^9|a^2=ef,b^2=e,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,e^d=ei,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, 326389);