Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,25,22,26,21,31)(19,27,28,20,29,24,30,32);(1,3)(2,6)(4,7)(5,8)(17,19)(18,24)(20,31)(21,30)(22,29)(23,28)(25,32)(26,27);(1,4,5,6)(2,3,7,8)(10,13)(14,15)(17,20,22,32)(18,19,26,29)(21,27,23,24)(25,28,31,30);(2,7)(3,8)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,25,26,31)(19,28,29,30)(20,27,32,24)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=f,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, 291969);