Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437999 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24);(9,10)(11,17,19,14)(12,20,21,15)(13,16)(18,22)(23,24);(2,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24);(11,19)(14,17)(18,24)(22,23)Polycyclic presentation:
2^9|b^2=f,d^2=g,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225694, #72225702, #72225703, … |
| maximal normal subgroup of | #72225694, #72225702, #72225703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356305);