Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499997 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,13)(12,19,20,24)(14,16)(15,21,17,23)(18,22);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,15,20,17)(14,16,22,18)(19,21,24,23);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,17)(13,20)(14,23)(16,24)(18,19)(21,22);(3,8)(5,7)(12,20)(15,17)(19,24)(21,23);(2,6)(5,7)(10,11)(15,17)(16,18)(19,24)Polycyclic presentation:
2^9|a^2=fi,b^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407196);