Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14375 |
| 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 | |
1060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,5,19)(3,8,13,24,14,22,20,23)(4,9,18,27,16,26,17,28)(12,21,15,31,29,32,30,25);(1,3)(2,8)(4,15)(5,20)(6,13)(7,14)(9,25)(10,22)(11,24)(12,17)(16,30)(18,29)(19,23)(21,27)(26,31)(28,32);(1,4)(2,9,10,26)(3,12)(5,18)(6,17)(7,16)(8,21,22,32)(11,28,19,27)(13,30)(14,29)(15,20)(23,31,24,25);(2,10)(5,6)(8,22)(9,26)(13,20)(15,30)(17,18)(21,32)Polycyclic presentation:
2^9|a^2=h,c^2=ef,g^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,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291450);