Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #398772 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2439 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14768 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
919 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)(3,12,14,7)(4,8)(5,9)(6,10)(11,19)(13,20,16,26)(15,22,28,21)(17,24)(18,23)(25,30)(27,31,29,32);(1,3)(2,7)(4,16)(5,15)(6,14)(8,20)(9,21)(10,12)(11,22)(13,17)(18,29)(19,28)(23,31)(24,26)(25,32)(27,30);(1,4)(2,8)(3,13,14,16)(5,18)(6,17)(7,20,12,26)(9,23)(10,24)(11,25)(15,27,28,29)(19,30)(21,31,22,32);(2,9,10,11)(7,21,12,22)(8,23,24,25)(20,31,26,32)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305968);