Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,11)(7,10,12,9)(13,14)(15,17)(16,18)(19,20)(21,22,24,25)(23,27,28,26);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20)(21,23,25,27,24,28,22,26);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(14,18)(17,20);(7,12)(9,10)(15,19)(17,20)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=g,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=defi,d^c=dei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299236);