Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25,21,24,31,32)(19,20,27,30,26,28,23,29);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19,21,26)(18,20,24,28)(22,27,31,23)(25,30,32,29);(1,4,6,5)(2,7,8,3)(10,14)(13,16)(17,20)(18,23)(19,25)(21,28)(22,30)(24,27)(26,32)(29,31);(1,5,6,4)(2,7,8,3)(10,14)(13,16)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291282);