Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394048 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,6)(3,7,8,5)(20,27)(21,29)(23,31)(24,26)(25,32)(28,30);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,31,30,32);(10,13)(14,16)(17,19,22,18)(20,25,28,23)(21,24,29,26)(27,31,30,32);(9,11,12,15)(10,14,13,16)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,31)(27,29)Polycyclic presentation:
2^9|a^2=e,b^2=ei,c^2=i,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311343);