Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#453 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,18,20,24)(13,16,22,17)(15,19,23,21);(1,3,5,4)(2,6,8,7)(10,14)(11,19)(13,22)(15,18)(20,21)(23,24);(9,11)(10,15)(12,20)(13,19)(14,23)(16,18)(17,24)(21,22);(10,16)(13,22)(14,17)(15,18)(19,21)(23,24);(1,4,5,3)(2,7,8,6)(10,14)(15,23)(16,17)(18,24)Polycyclic presentation:
2^9|a^2=f,b^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426504);