Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| 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 | |
1338 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)(6,9)(10,12)(13,14,16,18)(15,17,19,20)(21,22,25,29,26,30,23,27)(24,32,31,35,33,36,34,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,19)(18,20)(21,23,26,25)(22,27,30,29)(24,31,33,34)(28,32,35,36);(2,4)(5,7,8,11)(6,10,9,12)(15,19)(17,20);(6,9)(10,12)(21,24,26,33)(22,28,30,35)(23,31,25,34)(27,32,29,36)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358074);