Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
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 | |
1066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,3)(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,27)(22,26)(23,30)(24,28)(25,31)(29,32);(1,2,4,3)(5,7,8,6)(9,11,12,15)(10,13,14,16)(17,19,21,18)(20,25,29,23)(22,26,28,24)(27,30,32,31);(10,14)(11,15)(17,20,28,31,21,29,22,30)(18,23,26,32,19,25,24,27);(1,3,4,2)(18,19)(23,25)(24,26)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=egh,c^2=fgi,d^2=ei,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, 277586);