Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(9,10,13,14)(11,16,15,12)(17,18)(19,24)(20,23)(21,22);(2,4)(5,7,8,6)(10,14)(12,16)(17,19,22,20)(18,23,21,24);(1,2)(3,4)(10,14)(11,15)(19,20)(23,24);(9,11,13,15)(10,12,14,16)(17,20,22,19)(18,23,21,24);(9,12)(10,15)(11,14)(13,16)(17,21,22,18)(19,23,20,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 443190);