Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(5,6,8,7)(9,10,13,20,15,18,14,19)(11,21,25,32,23,31,24,16)(12,26,28,33,29,34,27,17)(22,30,37,35,36,40,38,39);(2,4)(9,11,14,24,15,23,13,25)(10,16,19,31,18,32,20,21)(12,22,27,38,29,36,28,37)(17,30,34,39,33,40,26,35);(5,7,8,6)(9,12,13,28,15,29,14,27)(10,17,20,26,18,33,19,34)(11,22,25,37,23,36,24,38)(16,30,21,35,32,40,31,39);(10,18)(11,23)(17,33)(19,20)(22,36)(24,25)(26,34)(37,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261158);