Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,18,22,25,21,24,20,26)(19,27,30,23,29,32,28,31);(1,2)(3,4)(10,13)(14,15)(17,19,21,29)(18,23,24,31)(20,30,22,28)(25,27,26,32);(2,4)(9,11,12,16)(10,14,13,15)(18,24)(23,31)(25,26)(27,32);(5,7,8,6)(10,13)(14,15)(18,25)(20,22)(23,27)(24,26)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fi,d^2=efhi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,h^b=hi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285629);