Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×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 | |
1546 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)(9,10,13,19,15,20,14,18)(11,21,24,16,23,32,25,31)(12,17,28,36,27,34,29,35)(22,37,38,30,26,40,39,33);(5,6,8,7)(9,11,14,25,15,23,13,24)(10,16,19,32,20,31,18,21)(12,26,29,38,27,22,28,39)(17,33,36,37,34,30,35,40);(2,4)(9,12)(10,17)(11,22)(13,28)(14,29)(15,27)(16,30)(18,35)(19,36)(20,34)(21,37)(23,26)(24,38)(25,39)(31,33)(32,40);(5,7,8,6)(11,23)(12,27)(16,31)(17,34)(21,32)(24,25)(28,29)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=di,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262128);