Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,12)(8,10)(9,11)(13,14,16,18)(15,19,20,17)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,36)(34,35);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,16,20)(14,17,18,19)(22,26)(27,35)(28,33)(29,36);(2,4)(5,8,9,7)(6,10,11,12)(14,18)(17,19);(21,23,24,30)(22,27,26,35)(25,34,32,31)(28,36,33,29)Polycyclic presentation:
2^9|a^2=f,b^2=ei,c^2=efi,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305897);