Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2540 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,21)(19,24,20,23);(6,8)(10,14)(12,15)(17,19,22,20)(18,23,21,24);(5,6,7,8)(17,18,22,21)(19,24,20,23);(1,2,3,4)(9,11)(10,15)(12,14)(13,16)(17,20,22,19)(18,23,21,24);(9,12,13,15)(10,11,14,16)(17,21,22,18)(19,23,20,24)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72265551, #72265558, #72265611, … |
| maximal normal subgroup of | #72265551, #72265558, #72265611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 582204);