Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432876 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4341 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)(3,5)(4,6)(7,8)(9,10)(11,20)(12,17)(13,18)(14,26)(15,19)(16,22)(21,36)(23,30)(24,32)(25,33)(27,31)(28,34)(29,38)(35,39)(37,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,21)(13,22)(14,27)(15,23)(17,29)(18,30)(19,31)(20,26)(24,37)(25,39)(28,38)(32,36)(33,40)(34,35);(3,7)(5,8)(9,12)(10,17)(11,21)(13,24)(14,28)(15,25)(16,29)(18,32)(19,33)(20,35)(22,37)(23,39)(26,34)(27,38)(30,36)(31,40);(2,6)(5,8)(9,13,15,14)(10,18,19,26)(11,22,23,27)(12,24,25,28)(16,30,31,20)(17,32,33,34)(21,37,39,38)(29,36,40,35);(10,19)(16,31)(17,33)(18,26)(20,30)(29,40)(32,34)(35,36)Polycyclic presentation:
2^9|d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285809, #72286231, #72286235, … |
| maximal normal subgroup of | #72285809, #72286231, #72286235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340074);