Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3154 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,17,12,18,11,16)(7,19,21,13,14,8,22,24)(9,25,26,15,28,31,27,32)(20,34,35,33,30,29,36,23);(1,2,4,3)(5,7,12,14)(6,13)(8,17)(9,20,28,30)(10,21,11,22)(15,29)(16,19)(18,24)(23,31)(25,33)(26,35,27,36)(32,34);(5,8,10,13,12,19,11,24)(6,14,16,22,18,7,17,21)(9,23,26,29,28,33,27,34)(15,30,25,36,32,20,31,35);(1,3,4,2)(5,9,10,26,12,28,11,27)(6,15,16,25,18,32,17,31)(7,20,21,35,14,30,22,36)(8,23,13,29,19,33,24,34)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276388);