Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14825 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,30)(3,14,18,34,19,33,17,9)(4,20,24,10,25,36,23,37)(5,26,28,39,29,11,27,38)(15,42,48,50,21,31,44,40)(16,45,49,32,47,41,46,56)(22,51,55,35,54,59,53,57)(43,61,63,64,52,58,62,60);(1,3,6,17,7,19,8,18)(2,9,12,14,13,34,30,33)(4,21,23,48,25,15,24,44)(5,16,27,46,29,47,28,49)(10,31,36,40,37,42,20,50)(11,32,38,41,26,56,39,45)(22,52,53,63,54,43,55,62)(35,58,59,60,57,61,51,64);(1,4)(2,10)(3,15)(5,22)(6,23)(7,25)(8,24)(9,31)(11,35)(12,36)(13,37)(14,40)(16,43)(17,44)(18,48)(19,21)(20,30)(26,57)(27,53)(28,55)(29,54)(32,58)(33,50)(34,42)(38,59)(39,51)(41,60)(45,64)(46,62)(47,52)(49,63)(56,61);(1,5,6,27,7,29,8,28)(2,11,12,38,13,26,30,39)(3,16,17,46,19,47,18,49)(4,22,23,53,25,54,24,55)(9,32,14,41,34,56,33,45)(10,35,36,59,37,57,20,51)(15,43,44,62,21,52,48,63)(31,58,40,60,42,61,50,64)Polycyclic presentation:
2^9|a^2=eg,b^2=e,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309138);