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 |
| prime radical | #61074 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(10,16)(11,21)(13,14)(18,31)(19,20)(23,35)(24,28)(25,34)(26,29)(30,33)(37,40)(38,39);(1,2)(3,7)(4,6)(5,8)(9,10)(11,22)(12,25)(13,16)(14,19)(15,20)(17,24)(18,26)(21,30)(23,37)(27,31)(28,33)(29,34)(32,35)(36,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,23)(13,28)(14,21)(15,24)(16,30)(18,32)(19,33)(20,22)(25,40)(26,38)(27,39)(29,35)(31,37)(34,36);(3,8)(5,7)(9,12,15,27)(10,18,20,34)(11,23,24,39)(13,26,14,29)(16,31,19,25)(17,32,22,36)(21,35,28,38)(30,37,33,40);(2,6)(5,7)(9,13,15,14)(10,19,20,16)(11,21,24,28)(12,26,27,29)(17,30,22,33)(18,25,34,31)(23,35,39,38)(32,37,36,40)Polycyclic presentation:
2^9|c^2=hi,d^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=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, 6962158);