Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| 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 | |
1238 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)(12,14)(13,16)(17,18,22,26,21,25,28,31)(19,27,29,24,20,23,30,32);(1,2,3,4)(9,11,12,16)(10,13,14,15)(17,19,28,30,21,20,22,29)(18,23,31,24,25,27,26,32);(2,4)(17,20,21,19)(18,24,25,32)(22,29,28,30)(23,26,27,31);(5,7,8,6)(10,14)(13,15)(18,25)(23,27)(24,32)(26,31)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=i,d^2=fhi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286365);