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×#423 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2090 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)(4,8)(5,7)(9,10,14,19,15,18,13,27)(11,20,22,16,23,32,28,31)(12,17,25,36,26,34,29,35)(21,37,38,30,39,33,24,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,24)(13,28)(14,22)(15,23)(17,33)(18,31)(19,32)(20,27)(21,25)(26,38)(29,39)(30,35)(34,37)(36,40);(1,4,5,3)(2,6,7,8);(9,12)(10,17)(11,21)(13,29)(14,25)(15,26)(16,30)(18,34)(19,36)(20,37)(22,38)(23,39)(24,28)(27,35)(31,40)(32,33);(10,18)(16,31)(17,34)(19,27)(20,32)(30,40)(33,37)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,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, 6287739);