Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1232 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)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,17,19,20)(22,26)(23,29)(24,31)(27,30)(28,34)(32,35);(6,9)(10,12)(13,15)(14,17)(16,19)(18,20)(21,22,24,26,25,28,31,34)(23,30,29,36,32,27,35,33);(5,7,8,11)(6,10,9,12)(15,19)(17,20)(21,23,31,35,25,32,24,29)(22,27,34,36,28,30,26,33);(2,4)(22,28)(26,34)(27,30)(33,36)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309166);