Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,21,20,24);(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,22)(23,24);(2,6)(7,8)(10,14)(11,16)(17,20,23,24,22,19,18,21);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,18,22,23)(19,21,20,24);(1,4,5,3)(2,8,6,7)(17,21,23,20,22,24,18,19)Polycyclic presentation:
2^9|a^2=i,c^2=fhi,d^2=fg,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319832 |
| maximal normal subgroup of | #72319832 |
| maximal quotient of | #72320116 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6749349);