Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #402003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2494 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14828 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,10)(3,13,17,25,16,8,15,24)(4,18,20,27,21,28,22,9)(14,29,31,26,19,32,30,23);(1,3,5,15,7,16,6,17)(2,8,12,25,11,13,10,24)(4,19,20,31,21,14,22,30)(9,26,18,29,27,23,28,32);(1,4)(2,9)(3,14)(5,22)(6,20)(7,21)(8,23)(10,18)(11,27)(12,28)(13,26)(15,31)(16,19)(17,30)(24,32)(25,29);(2,10,11,12)(8,24,13,25)(9,18,27,28)(23,32,26,29)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318217, #72318910, #72318991, … |
| maximal normal subgroup of | #72318217, #72318910, #72318991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309199);