Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500740 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,16)(12,25)(13,19)(14,30)(15,20)(17,27)(18,23)(21,35)(22,32)(24,33)(26,31)(28,37)(29,36)(34,38)(39,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,34)(18,30)(19,32)(20,33)(21,28)(25,31)(27,38)(29,39)(35,37)(36,40);(9,12,15,29)(10,17,20,37)(11,21,24,38)(13,27,14,28)(16,31,33,40)(18,35,32,34)(19,36,30,25)(22,26,23,39);(2,7)(3,8)(9,11)(10,18)(12,21)(13,23)(14,22)(15,24)(16,30)(17,35)(19,33)(20,32)(25,31)(26,28)(27,39)(29,38)(34,37)(36,40);(1,4,5,6)(2,3,7,8)(9,13,15,14)(10,19,20,30)(11,22,24,23)(12,27,29,28)(16,32,33,18)(17,36,37,25)(21,26,38,39)(31,34,40,35)Polycyclic presentation:
2^9|c^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409740);