Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(6,12,15,16)(7,17,19,24)(8,20)(14,29,30,21)(18,32,34,31)(22,23)(26,35)(33,36);(5,6,9,16,11,15,10,12)(7,13,17,27,19,28,24,25)(8,21,22,14,20,29,23,30)(18,33,31,26,34,36,32,35);(1,2,4,3)(5,7)(6,13)(8,18)(9,17)(10,24)(11,19)(12,25)(14,26)(15,28)(16,27)(20,34)(21,33)(22,31)(23,32)(29,36)(30,35);(1,3,4,2)(5,8,10,23,11,20,9,22)(6,14,16,29,15,30,12,21)(7,18,24,32,19,34,17,31)(13,26,27,36,28,35,25,33)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=efg,d^2=fg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264241);