Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395166 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,12)(8,11)(9,10)(13,14,19,15,18,24,20,23)(16,22,17,21,26,25,27,28);(2,4)(6,10)(11,12)(13,15,20,24,18,23,19,14)(16,25,27,28,26,22,17,21);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,20,27,18,26,19,17)(14,21,15,25,24,28,23,22);(5,8,9,7)(6,11,10,12)(13,17,19,26,18,27,20,16)(14,22,23,28,24,25,15,21)Polycyclic presentation:
2^9|a^2=efi,b^2=h,c^2=fgi,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317873);