Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,25)(21,31)(22,28)(23,27)(24,29)(30,32);(1,2)(3,4)(10,13)(14,15)(17,19,23,21,24,30,22,20)(18,25,28,32,29,31,27,26);(2,4)(17,20,23,19,24,21,22,30)(18,26,28,25,29,32,27,31);(9,11,12,16)(10,14,13,15)(17,21,23,30,24,20,22,19)(18,26,28,25,29,32,27,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 268129);