Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
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 | |
1204 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)(22,27)(23,31)(29,30)(33,34);(5,7,9,6)(8,11,12,10)(13,15,16,20)(14,17,18,19)(21,22,25,27)(23,32,31,36)(24,30,26,29)(28,34,35,33);(5,8,9,12)(6,10,7,11)(14,18)(17,19)(21,23,25,31)(22,28,27,35)(24,33,26,34)(29,32,30,36);(6,7)(10,11)(15,20)(17,19)(21,24,25,26)(22,29,27,30)(23,33,31,34)(28,32,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=efi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291473);