Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2543 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(5,6,7,8)(10,12)(14,15)(18,22)(20,23);(9,10,13,12)(11,14,16,15)(17,18)(19,20)(21,22)(23,24);(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,19,21,24)(18,23,22,20);(1,3,4,2)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23);(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=f,d^2=gi,e^2=fh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424314);