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 | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,13,10,14,8,11)(7,12,15,19,16,20,17,18)(21,22,26,29,25,28,34,24)(23,30,33,31,32,27,35,36);(1,2,3,4)(6,11)(8,9)(12,18)(13,14)(15,17)(19,20)(21,23)(22,27)(24,31)(25,32)(26,33)(28,30)(29,36)(34,35);(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,19)(14,20)(21,24,34,28,25,29,26,22)(23,31,35,30,32,36,33,27);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,18,20,19)(23,32)(27,30)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,d^2=h,e^2=h,f^2=i,b^a=be,d^a=dh,d^b=di,e^b=eh,f^b=fh,d^c=dh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 90840);