Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(5,6,7,9,8,10,11,12)(13,15)(14,18,20,21)(16,22)(17,23)(19,25,26,27)(24,28);(1,3,4,2)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28);(16,24)(19,26)(22,28)(25,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,g^2=i,b^a=be,g^a=gh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89140);