Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996395 |
| 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 | |
1753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,7)(9,10,16,21)(11,22,27,35)(12,28,30,40)(13,19,23,34)(14,20,15,33)(17,26,36,25)(18,29,37,24)(31,38,32,39);(1,2,5,6)(3,4,8,7)(9,11)(10,17)(12,23)(13,30)(14,25)(15,26)(16,27)(18,34)(19,37)(20,35)(21,36)(22,33)(24,31)(28,39)(29,32)(38,40);(1,3)(2,7)(4,6)(5,8)(9,12,15,24,16,30,14,29)(10,18,20,28,21,37,33,40)(11,23,26,31,27,13,25,32)(17,34,35,39,36,19,22,38);(2,6)(3,8)(9,13)(10,19)(11,24)(12,25)(14,31)(15,32)(16,23)(17,28)(18,22)(20,38)(21,34)(26,30)(27,29)(33,39)(35,37)(36,40);(1,4,5,7)(2,3,6,8)(9,14,16,15)(10,20,21,33)(11,25,27,26)(12,29,30,24)(13,31,23,32)(17,35,36,22)(18,28,37,40)(19,38,34,39)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092630);