Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | |
2086 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)(11,20)(12,17)(13,30)(14,19)(15,23)(16,27)(18,31)(21,33)(22,39)(24,35)(25,34)(26,32)(28,37)(29,36)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,26)(13,24)(14,25)(16,33)(17,38)(18,34)(19,35)(20,30)(21,29)(22,28)(23,31)(27,40)(32,37)(36,39);(1,3,4,8)(2,5,6,7)(9,12)(10,16)(11,21)(13,28)(14,29)(15,32)(17,30)(18,36)(19,37)(20,39)(22,23)(24,40)(25,26)(27,31)(33,35)(34,38);(3,8)(5,7)(9,12)(10,17)(11,22)(13,27)(14,29)(15,33)(16,30)(18,37)(19,36)(20,39)(21,23)(24,26)(25,40)(28,31)(32,35)(34,38);(2,6)(5,7)(9,13,14,31)(10,18,19,30)(11,23,25,24)(12,27,29,28)(15,20,35,34)(16,17,37,36)(21,40,26,22)(32,38,33,39)Polycyclic presentation:
2^9|c^2=hi,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=dhi,e^c=ei,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, 6962156);