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 | |
1932 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)(11,21,12,25)(13,27)(14,31)(15,19)(16,20)(17,24,34,23)(18,30)(22,39,26,35)(28,33,40,36)(29,37)(32,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,16)(13,28)(14,24)(15,23)(18,35)(19,25)(20,34)(21,31)(22,29)(26,32)(27,39)(30,40)(33,38)(36,37);(1,3)(2,5)(4,8)(6,7)(9,12,16,11)(10,17,20,34)(13,22,30,26)(14,24,15,23)(18,36,27,33)(19,21,31,25)(28,32,40,29)(35,38,39,37);(3,8)(5,7)(9,13,14,32,16,30,15,29)(10,18,31,37,20,27,19,38)(11,22,23,40,12,26,24,28)(17,33,25,39,34,36,21,35);(2,6)(5,7)(9,14,16,15)(10,19,20,31)(11,23,12,24)(13,29,30,32)(17,21,34,25)(18,37,27,38)(22,28,26,40)(33,39,36,35)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6914450);