Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1148 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,30)(24,31)(25,35)(26,29)(27,32)(28,33)(34,36);(5,7,8,11)(6,9,10,12)(13,15,16,20)(14,17,18,19)(21,23,24,25,27,34,26,33)(22,28,29,36,32,35,31,30);(6,10)(9,12)(15,20)(17,19)(21,24,27,26)(22,29,32,31)(23,33,34,25)(28,30,35,36);(14,18)(17,19)(21,25,24,34,27,33,26,23)(22,30,29,28,32,36,31,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=i,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270864);