Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | |
1908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,37,16,21,15,20)(11,22,25,17,27,38,26,33)(12,28,30,18,31,32,29,24)(13,19,34,40,36,23,35,39);(2,5)(3,7)(9,11)(10,17)(12,19)(13,32)(14,26)(15,25)(16,27)(18,34)(20,38)(21,33)(22,37)(23,31)(24,35)(28,36)(29,40)(30,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,30,16,31,15,29)(10,18,20,28,21,24,37,32)(11,23,26,40,27,19,25,39)(13,33,34,22,36,17,35,38);(2,5)(6,8)(9,13,14,34,16,36,15,35)(10,19,37,40,21,23,20,39)(11,24,25,28,27,18,26,32)(12,17,30,38,31,33,29,22);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,20,21,37)(11,25,27,26)(12,29,31,30)(13,34,36,35)(17,22,33,38)(18,32,24,28)(19,39,23,40)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7063402);