Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
778 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)(7,14)(8,10)(9,13)(11,15)(12,17)(16,19)(18,20)(21,22,26,29,25,28,34,36)(23,30,32,33,31,27,35,24);(1,2,4,3)(6,10)(8,17)(11,18)(12,13)(14,19)(16,20)(21,23)(22,27)(24,29)(25,31)(26,32)(28,30)(33,36)(34,35);(1,3,4,2)(5,7)(6,11)(8,16)(9,15)(10,18)(12,19)(13,14)(17,20)(22,28)(23,31)(29,36)(32,35);(7,15)(11,14)(16,20)(18,19)(21,24,26,30,25,33,34,27)(22,23,29,32,28,31,36,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,b^a=be,c^a=chi,e^a=eh,c^b=ci,e^b=eh,d^c=dh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91094);