Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(6,8)(9,10,12,11)(13,14)(15,18)(16,20)(17,19)(21,25)(22,27)(23,26)(24,28);(1,2,4,3)(6,8)(9,11,12,10)(13,15,16,22,17,23,24,21)(14,18,20,27,19,26,28,25);(1,2,4,3)(5,6,7,8)(15,21)(16,24)(18,25)(20,28)(22,23)(26,27);(1,3,4,2)(14,19)(18,26)(20,28)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274715);