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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1060 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,29)(24,28)(25,34)(26,31)(27,30)(32,35)(33,36);(6,9)(10,11)(13,15,16,20)(14,17,18,19)(21,23,25,33,26,24,27,32)(22,28,34,35,31,29,30,36);(14,18)(17,19)(21,24,26,23)(22,29,31,28)(25,33,27,32)(30,36,34,35);(5,7,8,12)(6,10,9,11)(14,18)(17,19)(21,25,26,27)(22,30,31,34)(23,32,24,33)(28,35,29,36)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=i,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269784);