Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
2040 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)(9,10,16,21)(11,22,26,28)(12,25,17,24)(13,18,33,37)(14,19,15,20)(23,39,35,34)(27,40,30,29)(31,36,32,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,21)(13,29)(14,24)(15,25)(16,26)(18,34)(19,22)(20,28)(23,32)(27,38)(30,36)(31,35)(33,40)(37,39);(1,3,4,8)(2,5,6,7)(9,12,16,17)(10,11,21,26)(13,30,33,27)(14,28,15,22)(18,35,37,23)(19,24,20,25)(29,36,40,38)(31,39,32,34);(3,8)(5,7)(9,13)(10,18)(11,23)(12,27)(14,31)(15,32)(16,33)(17,30)(19,36)(20,38)(21,37)(22,39)(24,29)(25,40)(26,35)(28,34);(2,6)(5,7)(9,14,16,15)(10,19,21,20)(11,24,26,25)(12,28,17,22)(13,31,33,32)(18,36,37,38)(23,29,35,40)(27,34,30,39)Polycyclic presentation:
2^9|a^2=i,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,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6962148);