Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,27)(20,22)(24,28)(29,30)(31,32);(9,11,12,16)(10,13,14,15)(17,18,21,23)(19,28,29,32)(20,25,22,26)(24,27,31,30);(1,2,3,4)(17,19,20,27,21,29,22,30)(18,24,26,32,23,31,25,28);(10,14)(13,15)(17,20,21,22)(18,25,23,26)(19,27,29,30)(24,28,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267816);