Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24);(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72252472, #72252475, #72252493, … |
| maximal normal subgroup of | #72252472, #72252475, #72252493, … |
| maximal quotient of | #72310611 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359461);