Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1082 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)(11,15)(13,16)(17,18)(19,26)(20,24)(21,23)(22,28)(25,30)(27,31)(29,32);(1,2,4,3)(5,7,8,6)(9,10,12,14)(11,16,15,13)(17,19)(18,22)(20,29)(21,27)(23,32)(24,26)(25,31)(28,30);(1,2,4,3)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(24,25)(28,29);(1,3,4,2)(18,24)(19,27)(20,30)(22,31)(23,25)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=fi,d^2=fg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280693);