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 | |
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)(13,16,31,36)(14,17,32,18)(21,37,38,33)(25,28,40,39)(29,34,30,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,12)(13,28)(14,22)(15,24)(16,33)(17,20)(18,26)(19,27)(21,30)(23,32)(25,34)(29,38)(31,39)(35,40)(36,37);(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)(26,32)(28,34)(29,37)(30,33)(31,40)(35,39)(36,38);(3,8)(5,7)(9,13)(10,16)(11,21)(12,25)(14,29)(15,31)(17,34)(18,35)(19,36)(20,37)(22,28)(23,39)(24,38)(26,33)(27,40)(30,32);(2,6)(5,7)(9,14,15,32)(10,17,19,18)(11,22,24,23)(12,20,27,26)(13,29,31,30)(16,34,36,35)(21,28,38,39)(25,37,40,33)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,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=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901078);