Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14526 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,28)(3,14,18,34,17,9,29,35)(4,19,22,10,23,38,30,37)(5,24,26,11,27,41,31,40)(15,43,46,32,20,51,45,42)(16,47,49,39,25,56,48,33)(21,36,54,59,55,60,53,50)(44,57,62,64,52,58,63,61);(1,3,6,29,8,17,7,18)(2,9,12,35,13,14,28,34)(4,20,22,45,23,15,30,46)(5,25,26,48,27,16,31,49)(10,32,19,43,37,42,38,51)(11,39,24,47,40,33,41,56)(21,44,53,63,55,52,54,62)(36,58,59,61,60,57,50,64);(1,4,7,22,8,23,6,30)(2,10,12,38,13,37,28,19)(3,15,18,46,17,20,29,45)(5,21,26,54,27,55,31,53)(9,32,35,51,14,42,34,43)(11,36,41,59,40,60,24,50)(16,44,49,62,25,52,48,63)(33,57,47,64,39,58,56,61);(1,5)(2,11)(3,16)(4,21)(6,31)(7,26)(8,27)(9,33)(10,36)(12,41)(13,40)(14,39)(15,44)(17,25)(18,49)(19,50)(20,52)(22,54)(23,55)(24,28)(29,48)(30,53)(32,57)(34,56)(35,47)(37,60)(38,59)(42,58)(43,61)(45,63)(46,62)(51,64)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295931);