Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220210 |
| 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 | |
1594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,16,10,15,27,14)(7,17,20,29,21,12,19,11)(8,22,25,13,26,32,24,33)(18,28,23,30,34,36,35,31);(2,4)(5,7)(6,11)(8,23)(9,20)(10,21)(12,14)(13,31)(15,29)(16,17)(18,24)(19,27)(22,36)(25,34)(26,35)(28,32)(30,33);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,21)(11,14)(13,30)(15,17)(16,29)(20,27)(22,36)(23,24)(25,35)(26,34)(28,32)(31,33);(5,8,9,24,10,26,27,25)(6,13,14,32,15,33,16,22)(7,18,19,23,21,34,20,35)(11,28,17,31,29,36,12,30)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293210);