Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1412 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)(6,10)(11,12)(14,17)(18,19)(21,22,24,28)(23,26,30,33)(25,27,32,29)(31,35,36,34);(5,6,9,10)(7,11,8,12)(13,14,16,17)(15,19,20,18)(21,23)(22,26)(24,30)(25,31)(27,35)(28,33)(29,34)(32,36);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,18,17,19)(23,30)(26,33)(31,36)(34,35);(5,8,9,7)(6,11,10,12)(14,17)(18,19)(22,27)(23,30)(25,32)(26,34)(28,29)(33,35)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=e,d^2=egi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355511);