Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432881 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62856 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10,13,21,14,17,12,16)(11,15,18,22,19,24,20,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,21)(14,17)(18,22)(19,24)(20,23);(2,5)(6,8)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,24)(21,23);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,18)(14,19)(16,23)(17,24)(21,22);(9,12,14,13)(10,16,17,21)(11,18,19,20)(15,22,24,23)Polycyclic presentation:
2^9|a^2=fgi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342612);