Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384334 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1022 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)(3,12,15,7)(4,16)(5,20)(6,10)(8,18)(9,14,24,21)(11,22)(13,25,17,30)(19,29)(23,28,27,31)(26,32);(1,3)(2,7)(4,17)(5,21)(6,15)(8,23)(9,11)(10,12)(13,19)(14,22)(16,25)(18,28)(20,24)(26,27)(29,30)(31,32);(1,4,5,18,6,19,22,32)(2,8,11,16,10,26,20,29)(3,13,14,28,15,17,21,31)(7,23,24,30,12,27,9,25);(1,3)(2,9,10,24)(4,13)(5,14)(6,15)(7,20,12,11)(8,25,26,30)(16,23,29,27)(17,19)(18,28)(21,22)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291530);