Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2826 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14648 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,19,33,20,41,18,9)(4,21,25,36,26,42,24,10)(5,11,29,38,30,40,28,39)(16,45,47,34,48,31,22,43)(17,44,51,59,52,58,50,32)(23,37,53,62,54,61,27,35)(46,63,64,60,55,56,49,57);(1,3,6,18,8,20,7,19)(2,9,12,15,14,33,13,41)(4,22,24,48,26,47,25,16)(5,17,28,50,30,52,29,51)(10,34,21,31,36,43,42,45)(11,32,38,44,40,59,39,58)(23,49,27,55,54,64,53,46)(35,60,37,56,62,57,61,63);(1,4,6,24,8,26,7,25)(2,10,12,21,14,36,13,42)(3,16,18,22,20,48,19,47)(5,27,29,23,30,53,28,54)(9,31,15,43,33,45,41,34)(11,37,39,35,40,61,38,62)(17,49,51,46,52,64,50,55)(32,57,58,56,59,60,44,63);(1,5,6,28,8,30,7,29)(2,11,13,39,14,40,12,38)(3,17,19,51,20,52,18,50)(4,23,25,53,26,54,24,27)(9,32,15,44,33,59,41,58)(10,35,21,37,36,62,42,61)(16,46,22,49,48,55,47,64)(31,56,34,60,45,63,43,57)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298830);