Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
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×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
836 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)(7,11)(9,12)(13,14)(15,20)(16,18)(17,19)(21,22)(23,30)(24,33)(25,29)(26,28)(27,32)(31,36)(34,35);(5,6)(7,9)(8,10)(11,12)(13,15,17,16)(14,18,19,20)(23,31)(24,26)(27,35)(28,33)(30,36)(32,34);(5,7,8,11)(6,9,10,12)(14,19)(15,16)(21,23,25,32)(22,27,29,30)(24,31,26,34)(28,36,33,35);(6,10)(9,12)(13,16,17,15)(14,18,19,20)(21,24,25,26)(22,28,29,33)(23,31,32,34)(27,36,30,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ef,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298631);