Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384326 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,12,6,8)(3,13,16,25,14,22,17,9)(4,18,20,26,21,10,19,23)(15,29,30,32,28,27,31,24);(2,8)(3,14)(4,19)(5,6)(9,22)(11,12)(13,25)(15,30)(20,21)(23,26)(27,29)(28,31);(1,3,6,17,7,14,5,16)(2,9,11,13,12,25,8,22)(4,15,20,30,21,28,19,31)(10,24,26,27,18,32,23,29);(1,4,5,19,7,21,6,20)(2,10,11,26,12,18,8,23)(3,15,16,31,14,28,17,30)(9,24,13,27,25,32,22,29)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251788, #72251789, #72251801, … |
| maximal normal subgroup of | #72251788, #72251789, #72251801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291524);