Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,28)(24,34)(25,35)(26,30)(27,32)(29,31)(33,36);(6,9)(10,12)(13,15,16,20)(14,17,18,19)(21,23,31,24,26,33,25,32)(22,27,35,36,30,34,29,28);(5,7,8,11)(6,10,9,12)(15,20)(17,19)(21,24,31,33,26,32,25,23)(22,28,35,27,30,36,29,34);(6,9)(10,12)(14,18)(17,19)(21,23,25,32,26,33,31,24)(22,27,29,28,30,34,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=eg,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270289);