Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| 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 | |
2132 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)(9,10,15,20,16,21,14,19)(11,22,25,34,27,37,26,17)(12,28,13,18,29,35,31,36)(23,39,40,33,24,32,30,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,23)(13,30)(14,25)(15,26)(16,27)(18,32)(19,22)(20,37)(21,34)(24,29)(28,38)(31,40)(33,35)(36,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,31,16,29,15,13)(10,18,20,35,21,36,19,28)(11,23,25,40,27,24,26,30)(17,32,37,33,34,39,22,38);(3,8)(5,7)(9,13,15,29,16,31,14,12)(10,18,20,35,21,36,19,28)(11,24,25,30,27,23,26,40)(17,33,22,32,34,38,37,39);(2,6)(5,7)(9,14,16,15)(10,19,21,20)(11,25,27,26)(12,13,29,31)(17,22,34,37)(18,35,36,28)(23,30,24,40)(32,33,39,38)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6928567);