Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2436 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14822 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1083 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,26)(3,14,27,32,18,39,17,9)(4,19,28,46,23,10,22,34)(5,11,29,35,25,36,24,52)(15,37,49,44,20,47,41,30)(16,38,53,56,43,59,42,31)(21,33,54,57,51,45,50,63)(40,60,64,55,48,58,61,62);(1,3,6,17,7,18,8,27)(2,9,12,14,13,32,26,39)(4,20,22,49,23,15,28,41)(5,16,24,42,25,43,29,53)(10,30,34,37,19,44,46,47)(11,31,35,38,36,56,52,59)(21,48,50,64,51,40,54,61)(33,55,57,58,45,62,63,60);(1,4,6,22,7,23,8,28)(2,10,12,34,13,19,26,46)(3,15,17,41,18,20,27,49)(5,21,24,50,25,51,29,54)(9,30,14,37,32,44,39,47)(11,33,35,57,36,45,52,63)(16,40,42,61,43,48,53,64)(31,55,38,58,56,62,59,60);(1,5)(2,11)(3,16)(4,21)(6,24)(7,25)(8,29)(9,31)(10,33)(12,35)(13,36)(14,38)(15,40)(17,42)(18,43)(19,45)(20,48)(22,50)(23,51)(26,52)(27,53)(28,54)(30,55)(32,56)(34,57)(37,58)(39,59)(41,61)(44,62)(46,63)(47,60)(49,64)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308901);