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 | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4350 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)(9,10,14,18)(11,15,21,27)(12,16,23,29)(13,17,24,30)(19,25,32,37)(20,26,33,38)(22,34,35,28)(31,39,40,36);(2,5)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,25)(17,26)(18,27)(22,31)(23,32)(24,33)(28,36)(29,37)(30,38)(34,39)(35,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,25)(17,28)(18,29)(20,31)(21,32)(24,35)(26,36)(27,37)(30,34)(33,40)(38,39);(9,13)(10,17)(11,20)(12,22)(14,24)(15,26)(16,28)(18,30)(19,31)(21,33)(23,35)(25,36)(27,38)(29,34)(32,40)(37,39);(11,21)(15,27)(19,32)(20,33)(25,37)(26,38)(31,40)(36,39)Polycyclic presentation:
2^9|a^2=i,c^2=f,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 | #72284555, #72284882, #72284967, … |
| maximal normal subgroup of | #72284555, #72284882, #72284967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536459);