Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682500 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #31224 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,13)(11,14)(15,16)(17,18)(19,24)(20,22)(21,23);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,15,16,14);(2,5)(6,8)(9,11,10,14,12,16,13,15)(17,19,24,18,21,22,20,23);(1,3,4,7)(2,6,5,8)(9,10,12,13)(11,14,16,15);(2,5)(6,8)(11,16)(14,15)(17,20,21,24)(18,22,23,19)Polycyclic presentation:
2^9|b^2=f,c^2=fgi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6695860);