Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993875 |
| 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 | |
1781 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,19,16,20,14,31)(11,21,23,17,12,24,26,33)(13,18,29,36,30,38,28,37)(22,39,40,32,25,34,27,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,16)(13,27)(14,26)(15,23)(18,34)(19,24)(20,33)(21,31)(22,29)(25,28)(30,40)(32,37)(35,36)(38,39);(1,3,4,8)(2,5,6,7)(9,12,16,11)(10,17,20,33)(13,22,30,25)(14,26,15,23)(18,35,38,32)(19,21,31,24)(27,28,40,29)(34,37,39,36);(3,8)(5,7)(9,13,16,30)(10,18,20,38)(11,22,12,25)(14,28,15,29)(17,32,33,35)(19,36,31,37)(21,39,24,34)(23,40,26,27);(2,6)(5,7)(9,14,16,15)(10,19,20,31)(11,23,12,26)(13,28,30,29)(17,21,33,24)(18,36,38,37)(22,40,25,27)(32,39,35,34)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970286);