Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196038 |
| 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, cosocle | C23×C4 | C24 |
| prime radical | C2×#2404 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,9,11,12)(8,10)(13,14,17,21,18,20,25,15)(16,19,24,27,23,26,28,22);(5,7)(6,9,10,12)(8,11)(13,15,17,20,18,21,25,14)(16,22,24,26,23,27,28,19);(2,4)(13,16)(14,19)(15,22)(17,24)(18,23)(20,26)(21,27)(25,28);(16,23)(19,26)(22,27)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=eg,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,g^b=gh,d^c=di86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103234);