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 | |
1808 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)(9,10,15,18,14,20,13,19)(11,21,24,22,26,33,25,16)(12,17,30,35,31,28,29,36)(23,37,27,38,40,34,39,32);(1,2)(3,7)(4,6)(5,8)(9,11,13,25,14,26,15,24)(10,16,18,21,20,22,19,33)(12,27,30,40,31,39,29,23)(17,34,36,38,28,37,35,32);(1,3)(2,5)(4,8)(6,7)(9,10,14,20)(11,22,26,16)(12,28,31,17)(13,18,15,19)(21,25,33,24)(23,32,40,38)(27,34,39,37)(29,36,30,35);(3,8)(5,7)(9,12)(10,17)(11,23)(13,29)(14,31)(15,30)(16,32)(18,35)(19,36)(20,28)(21,37)(22,38)(24,27)(25,39)(26,40)(33,34);(2,6)(5,7)(9,13,14,15)(10,18,20,19)(11,24,26,25)(12,29,31,30)(16,33,22,21)(17,35,28,36)(23,27,40,39)(32,34,38,37)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^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,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, 6910261);