Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#432 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,18,13,28)(11,20,22,16,23,33,29,32)(12,24,26,35,27,17,30,36)(21,34,38,40,39,37,25,31);(9,11)(10,16)(12,25)(13,29)(14,22)(15,23)(17,34)(18,32)(19,33)(20,28)(21,26)(24,37)(27,38)(30,39)(31,35)(36,40);(2,5)(6,8);(9,12)(10,17)(11,21)(13,30)(14,26)(15,27)(16,31)(18,24)(19,36)(20,37)(22,38)(23,39)(25,29)(28,35)(32,40)(33,34);(1,3,4,7)(2,6,5,8)(10,18)(16,32)(17,24)(19,28)(20,33)(31,40)(34,37)(35,36)Polycyclic presentation:
2^9|a^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,d^b=df,f^b=fi,f^d=fi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288185);