Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18,20,21)(19,22,23,24);(1,2)(3,4)(9,11)(10,13)(12,15)(14,16)(18,21)(22,24);(2,4)(10,14)(13,16)(17,18)(19,22)(20,21)(23,24);(6,8)(11,15)(13,16)(18,21)(22,24);(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^9|a^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72303410, #72303412, #72304055, … |
| maximal normal subgroup of | #72303410, #72303412, #72304055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 752058);