Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384777 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2337 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14477 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,13,6,12)(3,15,19,9,20,34,18,35)(4,21,26,38,25,10,24,39)(5,27,29,11,30,41,31,42)(16,43,45,32,46,36,22,51)(17,47,28,59,48,33,49,40)(23,53,55,50,56,58,57,37)(44,63,52,61,54,60,64,62);(1,3,8,18,7,20,6,19)(2,9,14,15,13,35,12,34)(4,22,26,46,25,45,24,16)(5,28,31,48,30,49,29,17)(10,36,39,32,21,43,38,51)(11,40,27,47,42,59,41,33)(23,54,57,64,56,44,55,52)(37,60,58,62,50,63,53,61);(1,4,6,24,7,25,8,26)(2,10,12,38,13,21,14,39)(3,16,18,22,20,46,19,45)(5,23,29,55,30,56,31,57)(9,32,15,43,35,51,34,36)(11,37,41,53,42,50,27,58)(17,44,28,52,48,54,49,64)(33,60,40,62,47,63,59,61);(1,5,6,29,7,30,8,31)(2,11,14,27,13,42,12,41)(3,17,18,28,20,48,19,49)(4,23,24,55,25,56,26,57)(9,33,34,59,35,47,15,40)(10,37,39,58,21,50,38,53)(16,44,22,52,46,54,45,64)(32,60,36,61,51,63,43,62)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294732);