Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1438 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)(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,17,19,20)(21,22)(23,27)(24,28)(25,29)(26,31)(30,34)(32,35)(33,36);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,24)(22,27,31,28)(25,32,34,33)(29,35,30,36);(2,4)(15,19)(17,20)(21,24,26,23)(22,28,31,27)(25,33,34,32)(29,36,30,35);(5,8,9,7)(6,10,11,12)(22,29)(25,34)(27,35)(28,36)(30,31)(32,33)Polycyclic presentation:
2^9|a^2=eg,b^2=ei,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309089);