Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #272667 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #2856 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1031 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,21)(19,22,24,23);(2,5)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,20,24)(18,22,21,23);(9,10)(11,15,16,13)(12,14)(18,21)(19,23)(22,24);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=ei,b^2=i,c^2=fi,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gi,g^c=ghi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 202506);