Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500015 |
| 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 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,22)(20,21);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(3,8)(5,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24);(2,6)(5,7)(12,21)(15,20)(18,24)(22,23)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=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319306, #72319407 |
| maximal normal subgroup of | #72319306, #72319407 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407219);