Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,7)(9,10)(11,12)(13,14,18,21,17,15,19,23)(16,20,25,27,24,22,26,28);(5,6,8,7)(13,15,18,23,17,14,19,21)(16,22,25,28,24,20,26,27);(2,4)(5,7,8,6)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28);(10,12)(14,15)(16,24)(21,23)(25,26)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dg,d^b=di,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300027);