Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6346762 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(1,3)(2,5)(4,7)(6,8)(12,13)(16,21)(18,23)(19,22);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18);(10,17)(11,20)(16,21)(19,22)Polycyclic presentation:
2^9|d^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72234732 |
| maximal normal subgroup of | #72234732 |
| maximal quotient of | #72234738, #72284163, #72284282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253959);