Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455107 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,5)(3,8,4,7)(9,10,12,14)(11,13,15,16)(17,18,22,25)(19,24,31,28)(20,21,26,23)(27,29,32,30);(1,3,6,4)(2,7,5,8)(9,11,12,15)(10,13,14,16)(17,18,21,20,22,25,23,26)(19,27,29,28,31,32,30,24);(1,4,6,3)(2,7,5,8)(11,15)(13,16)(17,19,23,29,22,31,21,30)(18,24,26,27,25,28,20,32);(1,5,6,2)(3,8,4,7)(17,20,23,18,22,26,21,25)(19,28,30,27,31,24,29,32)Polycyclic presentation:
2^9|a^2=eg,b^2=egh,c^2=ef,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381357);