Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
920 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,9)(8,16)(10,13)(11,14)(12,18)(15,19)(17,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(2,4)(5,7,9,14,11,13,10,6)(8,15,18,20,19,16,17,12)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,8)(6,12)(7,15)(9,18)(10,17)(11,19)(13,16)(14,20)(23,29)(26,32)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ci,d^b=dh,e^b=eh,d^c=dh,e^c=eh128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12869);