Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623305 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1685 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)(3,7)(4,6)(5,8)(9,10,15,20)(11,21,24,38)(12,19,14,17)(13,26,29,34)(16,31,33,23)(18,25,36,28)(22,39,30,35)(27,37,40,32);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,11)(10,16)(12,22)(13,27)(14,30)(15,24)(17,32)(18,35)(19,37)(20,33)(21,26)(23,28)(25,31)(29,40)(34,38)(36,39);(2,6)(5,7);(9,12,15,14)(10,17,20,19)(11,22,24,30)(13,25,29,28)(16,32,33,37)(18,34,36,26)(21,35,38,39)(23,27,31,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gh,h^a=hi,c^b=cf,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530555);