Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2430 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4013 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)(10,11)(12,16)(19,23)(22,24);(9,10,13,11)(12,14,16,15)(18,21)(22,24);(5,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,21)(23,24);(1,2)(3,4)(17,19,20,23)(18,22,21,24);(9,12)(10,14)(11,15)(13,16)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
| maximal subgroup of | #72280221, #72280272, #72280503, … |
| maximal normal subgroup of | #72280221, #72280272, #72280503, … |
| maximal quotient of | C2×#613224 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 520420);