Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,11)(7,10,8,12)(13,14,16,18)(15,17,19,20)(21,22)(23,29)(24,32)(25,27)(26,30)(28,34)(31,36)(33,35);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,24,30,25,31,34,33)(22,26,28,29,27,35,32,36);(5,8,9,7)(6,10,11,12)(15,19)(17,20)(22,27)(26,35)(28,32)(29,36);(2,4)(21,23,25,31)(22,26,27,35)(24,33,34,30)(28,36,32,29)Polycyclic presentation:
2^9|a^2=fgi,b^2=efi,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=dei,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342654);