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) = C28 |
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 | |
3486 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)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,23)(19,24,20,21);(9,10,12,14)(11,13,16,15)(18,23)(19,20);(1,2,3,4)(17,19,22,20)(18,21,23,24);(9,10)(11,15)(12,14)(13,16)(17,20,22,19)(18,21,23,24);(9,11)(10,13)(12,16)(14,15)(17,21)(18,20)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=g,d^2=h,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, 426432);