Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184773 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,23,24,19)(17,20)(21,27,28,25)(22,26);(1,2,4,3)(5,7,8,11)(6,9,10,12);(6,10)(9,12)(13,15,17,22)(14,18,20,26)(16,21,24,28)(19,25,23,27);(1,3,4,2)(13,16)(14,19)(15,21)(17,24)(18,25)(20,23)(22,28)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=i,d^2=e,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299582);