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 | C24 | C2×C22⋊C4 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,17,19,20)(16,18)(21,22,23,25,24,26,28,27);(5,7,11,6,9,12,8,10)(13,15,16,19)(14,17)(18,20)(22,25,26,27);(1,2,3,4)(6,10)(7,12)(14,18)(15,19)(22,26)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cgi,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, 9180);