Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
1895 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)(9,10,16,22)(11,23,12,26)(13,29,33,19)(14,21,15,20)(17,28,18,25)(24,36,27,39)(30,34,40,35)(31,37,32,38);(2,5)(3,7)(9,11)(10,17)(12,16)(13,30)(14,28)(15,25)(18,22)(19,36)(20,26)(21,23)(24,32)(27,31)(29,39)(33,40)(34,38)(35,37);(1,2)(3,6)(4,5)(7,8)(9,12,16,11)(10,18,22,17)(13,24,33,27)(14,28,15,25)(19,34,29,35)(20,26,21,23)(30,31,40,32)(36,37,39,38);(2,5)(6,8)(9,13)(10,19)(11,24)(12,27)(14,31)(15,32)(16,33)(17,34)(18,35)(20,37)(21,38)(22,29)(23,39)(25,40)(26,36)(28,30);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,20,22,21)(11,25,12,28)(13,31,33,32)(17,23,18,26)(19,37,29,38)(24,40,27,30)(34,39,35,36)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050016);