Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| 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 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(6,8)(13,14,19,15,18,22,17,21)(16,20,25,23,26,28,27,24);(1,2,4,3)(9,10,11,12)(13,15,19,22,18,21,17,14)(16,24,25,20,26,23,27,28);(5,6,7,8)(10,12)(13,16)(14,20)(15,23)(17,27)(18,26)(19,25)(21,24)(22,28);(1,3,4,2)(13,17,18,19)(14,21,22,15)(16,25,26,27)(20,23,28,24)Polycyclic presentation:
2^9|a^2=h,b^2=egh,c^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72257834, #72257842, #72257848 |
| maximal normal subgroup of | #72257834, #72257842, #72257848 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274461);