Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×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 | |
928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,13)(7,10,15,9)(8,12,17,20)(14,16,19,18)(21,22,25,29,26,28,34,35)(23,27,31,24,30,32,36,33);(2,4)(5,7,10,13,11,15,9,6)(8,14,16,20,17,19,18,12)(21,23)(22,27)(24,29)(25,31)(26,30)(28,32)(33,35)(34,36);(5,8)(6,12)(7,14)(9,18)(10,16)(11,17)(13,20)(15,19)(21,24,25,32,26,33,34,27)(22,23,29,31,28,30,35,36);(5,9,11,10)(6,7,13,15)(8,16,17,18)(12,19,20,14)(22,28)(23,30)(29,35)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=ci,d^a=di,e^a=eh,g^a=gh,d^b=dhi,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89759);