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 | |
1174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,17,19,20)(21,22,25,28)(23,29,31,36)(24,32,34,27)(26,33,35,30);(1,2,4,3)(6,9)(10,12)(13,15)(14,17)(16,19)(18,20);(1,3,4,2)(5,7,8,11)(6,10,9,12)(15,19)(17,20);(21,23,25,31)(22,26,28,35)(24,33,34,30)(27,36,32,29)Polycyclic presentation:
2^9|a^2=efgi,b^2=f,c^2=e,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, 309147);