Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1624 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)(9,10,12,11)(13,14)(15,23)(16,25)(17,21)(18,26)(19,20)(22,27)(24,28);(1,2,4,3)(5,6)(7,8)(13,15)(14,19)(16,26)(17,20)(18,24)(21,23)(22,28)(25,27);(6,8)(13,16,27,23,17,28,18,19)(14,20,26,24,21,15,22,25);(1,3,4,2)(9,11,12,10)(14,21)(15,20)(22,26)(24,25)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=fi,d^2=ehi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284254);