Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15)(7,16,9,19)(8,13,10,20)(12,18,14,17)(21,22,26,30)(23,31,34,35)(24,27,25,29)(28,32,36,33);(1,2)(3,4)(5,7,11,9)(6,12,15,14)(8,17,10,18)(13,19,20,16)(22,27)(24,25)(28,35)(29,30)(31,36)(32,33);(2,4)(5,8,11,10)(6,13,15,20)(7,17,9,18)(12,19,14,16)(21,23,26,34)(22,28,30,36)(24,33,25,32)(27,35,29,31);(5,9,11,7)(6,14,15,12)(8,18,10,17)(13,16,20,19)(22,27)(24,25)(28,35)(29,30)(31,36)(32,33)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fi,e^c=ei,f^c=fi,e^d=eh,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 153686);