Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,22,27,31)(23,28,26,35)(24,33,34,30)(25,29,32,36);(6,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23,27,26)(22,28,31,35)(24,25,34,32)(29,30,36,33);(5,7,8,11)(6,10,9,12)(15,19)(17,20)(21,24,23,32,27,34,26,25)(22,29,35,30,31,36,28,33);(2,4)(7,11)(10,12)(21,25,26,34,27,32,23,24)(22,30,28,29,31,33,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=efi,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375158);