Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1745 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)(3,5)(4,6)(7,8)(9,10)(11,23,27,36)(12,28,31,38)(13,19)(14,20)(15,21)(16,22)(17,25,37,26)(18,29,35,30)(24,39)(32,40)(33,34);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,24)(13,30)(14,25)(15,26)(16,27)(18,34)(19,28)(20,23)(21,36)(22,37)(29,32)(31,33)(35,39)(38,40);(1,3,4,8)(2,5,6,7)(9,12,16,31)(10,18,22,35)(11,24,27,33)(13,25,32,26)(14,29,15,30)(17,34,37,39)(19,36,40,23)(20,28,21,38);(3,8)(5,7)(9,13,15,24,16,32,14,33)(10,19,21,39,22,40,20,34)(11,12,25,30,27,31,26,29)(17,35,23,28,37,18,36,38);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,25,27,26)(12,29,31,30)(13,24,32,33)(17,23,37,36)(18,28,35,38)(19,39,40,34)Polycyclic presentation:
2^9|a^2=fgi,c^2=hi,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972424);