Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500019 |
| 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 | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,14)(12,19,20,15)(13,16)(17,21,23,24)(18,22);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,24)(20,23);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,23)(22,24);(3,8)(5,7)(11,18)(14,22)(17,23)(21,24);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72226638, #72232592, #72234655, … |
| maximal normal subgroup of | #72226638, #72232592, #72234655, … |
| maximal quotient of | #72267781, #72267822, #72267863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407215);