Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #96924 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×C22⋊C4 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(14,17,18,19)(21,22,25,27,24,23,26,28);(5,6,7,11,9,12,8,10)(13,14,15,19,16,18,20,17)(21,23,25,28,24,22,26,27);(2,4)(5,7,9,8)(6,11,12,10)(22,23)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,d^b=dh,f^b=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6956);