Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,30)(24,27)(25,28)(26,29)(31,36)(32,35)(33,34);(1,2)(3,4)(6,9)(7,11)(13,15)(14,17)(16,19)(18,20)(21,23,26,31,25,24,32,33)(22,27,29,34,28,30,35,36);(2,4)(7,11)(10,12)(14,18)(17,20)(21,24)(22,27)(23,25)(26,31)(28,30)(29,36)(32,33)(34,35);(5,7,8,11)(6,10,9,12)(15,19)(17,20)(22,28)(23,24)(29,35)(31,33)Polycyclic presentation:
2^9|b^2=h,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,d^c=de,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298972);