Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196101 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,20,15,18,13,19)(11,16,22,32,23,31,29,36)(12,24,27,34,28,17,26,35)(21,37,25,40,38,30,39,33);(2,5)(6,8)(9,11)(10,16)(12,25)(13,29)(14,22)(15,23)(17,33)(18,31)(19,36)(20,32)(21,26)(24,40)(27,38)(28,39)(30,34)(35,37);(1,3)(2,6)(4,7)(5,8)(9,12,15,28)(10,17,18,24)(11,21,23,38)(13,27,14,26)(16,30,31,37)(19,35,20,34)(22,39,29,25)(32,40,36,33);(3,7)(6,8)Polycyclic presentation:
2^9|a^2=h,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286975);