Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,7,8)(10,12)(14,15)(17,18,21,22)(19,23,24,20);(1,2,3,4)(9,10,13,12)(11,15,16,14)(18,22)(19,24);(1,2)(3,4)(17,19)(18,20)(21,24)(22,23);(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,22)(21,24);(9,12,13,10)(11,14,16,15)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^9|a^2=i,b^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 426264);