Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,9)(8,11)(10,12)(14,17)(15,19)(21,22,24,28,25,27,31,34)(23,26,30,32,29,33,35,36);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14,16,17)(15,18,19,20);(2,4)(5,8,10,9)(6,11,12,7)(13,15,16,19)(14,18,17,20)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,9,10,8)(6,11,12,7)(14,17)(18,20)(22,27)(23,29)(28,34)(30,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292285);