Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1301 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,23)(21,26)(22,25)(24,29)(28,32)(30,31);(1,2)(3,7)(4,8)(5,6)(11,15)(13,16)(18,23)(19,28)(25,26)(29,30);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,31)(25,32)(26,27);(1,4,5,3)(2,7,6,8)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32)Polycyclic presentation:
2^9|a^2=gi,c^2=eg,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351827);