Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470028 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4229 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)(14,27)(15,19)(16,22)(21,36)(23,30)(24,32)(25,33)(26,34)(28,31)(29,37)(35,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,37)(25,38)(26,39)(32,35)(33,36)(34,40);(3,7)(5,8)(9,12)(10,17)(11,21)(13,25)(14,24)(15,26)(16,29)(18,33)(19,34)(20,35)(22,38)(23,39)(27,32)(28,37)(30,36)(31,40);(3,7)(5,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,26,25)(16,30,31,20)(17,32,34,33)(21,37,39,38)(29,35,40,36);(2,6)(5,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,25,26,24)(16,30,31,20)(17,33,34,32)(21,38,39,37)(29,36,40,35)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317526, #72317532, #72317536, … |
| maximal normal subgroup of | #72317526, #72317532, #72317536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377232);