Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184786 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,19,20,18,23,15,24)(16,22,26,28,17,21,25,27);(1,2,4,3)(6,9)(10,12)(13,15,18,19)(14,20,23,24)(16,26,17,25)(21,28,22,27);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,21,23,22)(15,25,19,26)(20,27,24,28);(13,17,18,16)(14,22,23,21)(15,25,19,26)(20,27,24,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299484);