Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628896 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #35659 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(19,31)(26,37)(32,39)(38,40);(2,5)(6,8)(9,10,14,18)(11,15,21,28)(12,16,23,30)(13,24,25,17)(19,26,31,37)(20,33,34,27)(22,35,36,29)(32,40,39,38);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,26)(17,27)(18,28)(22,32)(23,31)(24,33)(25,34)(29,38)(30,37)(35,40)(36,39);(9,12)(10,16)(11,19)(13,22)(14,23)(15,26)(17,29)(18,30)(20,32)(21,31)(24,35)(25,36)(27,38)(28,37)(33,40)(34,39);(9,13,14,25)(10,17,18,24)(11,20,21,34)(12,22,23,36)(15,27,28,33)(16,29,30,35)(19,32,31,39)(26,38,37,40)Polycyclic presentation:
2^9|b^2=i,c^2=f,e^2=i,b^a=bf,c^a=cg,d^a=dh,e^b=ei,h^c=hi,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72306350, #72306421, #72306431, … |
| maximal normal subgroup of | #72306350, #72306421, #72306431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536461);