Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437621 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4046 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,29)(22,30)(23,31)(24,32)(25,34)(26,35)(27,37)(28,38)(33,40)(36,39);(9,11)(10,16)(12,21)(13,22)(14,28)(15,24)(17,33)(18,36)(19,38)(20,32)(23,25)(26,40)(27,39)(29,35)(30,37)(31,34);(1,3,5,4)(2,6,7,8)(9,12,15,26)(10,17,20,35)(11,21,24,40)(13,25,27,14)(16,29,32,33)(18,34,37,19)(22,23,39,28)(30,31,36,38);(1,4,5,3)(2,6,7,8)(9,13)(10,18)(11,22)(12,25)(14,26)(15,27)(16,30)(17,34)(19,35)(20,37)(21,23)(24,39)(28,40)(29,31)(32,36)(33,38);(9,14,15,25)(10,19,20,34)(11,23,24,28)(12,13,26,27)(16,31,32,38)(17,18,35,37)(21,39,40,22)(29,36,33,30)Polycyclic presentation:
2^9|c^2=gi,d^2=g,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284218, #72284505, #72284696, … |
| maximal normal subgroup of | #72284218, #72284505, #72284696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359074);