Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6346647 |
| 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 | |
4052 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)(11,15)(12,14)(13,16)(17,20)(18,22)(19,21)(23,24);(1,2)(3,5)(4,6)(7,8)(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,18)(14,21)(16,22)(17,23)(20,24);(11,18)(15,22)(19,24)(21,23);(10,16)(14,17)(15,22)(21,23)Polycyclic presentation:
2^9|f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231835, #72236705, #72238132, … |
| maximal normal subgroup of | #72231835, #72236705, #72238132, … |
| maximal quotient of | #72231830, #72278134, #72285303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253843);