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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2080 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)(11,20,24,26)(12,22,27,23)(13,28,32,16)(14,17,33,18)(21,34,38,37)(25,39,40,29)(30,36,31,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,12)(13,29)(14,22)(15,24)(16,34)(17,20)(18,26)(19,27)(21,31)(23,33)(25,35)(28,37)(30,38)(32,39)(36,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,25)(14,20)(15,27)(16,21)(17,22)(18,23)(19,24)(26,33)(28,38)(29,35)(30,37)(31,34)(32,40)(36,39);(3,8)(5,7)(9,13)(10,16)(11,21)(12,25)(14,30)(15,32)(17,35)(18,36)(19,28)(20,37)(22,29)(23,39)(24,38)(26,34)(27,40)(31,33);(2,6)(5,7)(9,14,15,33)(10,17,19,18)(11,22,24,23)(12,20,27,26)(13,30,32,31)(16,35,28,36)(21,29,38,39)(25,37,40,34)Polycyclic presentation:
2^9|a^2=i,e^2=i,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, 6901090);