Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298711 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,13)(8,17)(9,11)(10,15)(12,14)(16,19)(18,20)(21,22)(23,30)(24,28)(25,34)(26,29)(27,31)(32,35)(33,36);(1,2)(3,4)(5,7,14,20,9,16,8,15)(6,10,17,19,11,18,12,13);(2,4)(6,11)(10,18)(12,17)(13,19)(21,23,25,31,26,32,24,33)(22,27,28,30,29,36,34,35);(5,7,14,20,9,16,8,15)(6,10,17,19,11,18,12,13)(21,24,26,25)(22,28,29,34)(23,31,32,33)(27,35,36,30)Polycyclic presentation:
2^9|b^2=eh,c^2=fh,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 205955);