Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500474 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,14)(11,18)(12,15)(17,22)(19,20)(21,23);(1,2)(3,7)(4,6)(5,8)(9,10,15,22,13,17,12,14)(11,16,18,23,20,24,19,21);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,24)(22,23);(3,8)(5,7);(2,6)(5,7)Polycyclic presentation:
2^9|b^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72213852, #72219565, #72219567 |
| maximal normal subgroup of | #72213852, #72219565, #72219567 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407672);