Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361285 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2290 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13900 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13,8,12,6,11)(3,14,4,18,17,25,16,9)(5,20,21,26,22,27,23,10)(15,29,30,28,31,24,19,32);(1,3)(2,9,12,18)(4,6)(5,15)(7,16)(8,17)(10,24,26,29)(11,14,13,25)(19,21)(20,28,27,32)(22,31)(23,30);(1,4,6,17,8,16,7,3)(2,9,11,14,12,18,13,25)(5,19,21,31,22,30,23,15)(10,24,20,28,26,29,27,32);(1,5,6,21,8,22,7,23)(2,10,13,27,12,26,11,20)(3,15,4,19,17,31,16,30)(9,24,25,32,18,29,14,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268734);