Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500482 |
| 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 | |
3149 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)(11,17)(12,21)(16,23)(19,20)(22,24);(1,2)(3,7)(4,6)(5,8)(9,10,12,14,13,16,21,23)(11,18,20,24,19,15,17,22);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,19)(14,22)(16,18)(17,21)(23,24);(3,8)(5,7)(10,16)(11,19)(14,23)(17,20);(2,6)(5,7)(10,16)(14,23)(15,18)(22,24)Polycyclic presentation:
2^9|b^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213852, #72219569 |
| maximal normal subgroup of | #72213852, #72219569 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407678);