Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| 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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1150 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)(8,12)(9,10)(13,14)(15,19)(16,17)(18,20)(21,22,24,29,26,30,25,28)(23,27,31,34,33,36,32,35);(1,2)(3,4)(6,10)(11,12)(14,17)(18,19)(21,23,24,31,26,33,25,32)(22,27,29,34,30,36,28,35);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,18,17,19)(22,28)(23,31)(24,25)(29,30)(32,33)(34,35);(5,8,9,7)(6,11,10,12)(14,17)(18,19)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=e,d^2=efg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280002);