Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60429 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1843 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,6,4)(9,10)(11,16)(12,14)(13,15)(17,20)(18,21)(19,22)(23,24);(1,3,4,2,5,6,8,7)(10,14)(15,17)(16,21)(22,23);(10,15)(13,20)(14,17)(16,22)(19,24)(21,23);(9,11)(10,16)(12,18)(13,19)(14,21)(15,22)(17,23)(20,24);(11,18)(16,21)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=fi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,g^c=gi,e^d=ei95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263206);