Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4174 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)(9,10,15,20)(11,21,23,33)(12,25,27,34)(13,19,14,18)(16,28,31,24)(17,29,32,26)(22,30,36,40)(35,37,39,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,22)(13,28)(14,24)(15,23)(17,30)(18,33)(19,21)(20,31)(25,39)(26,38)(27,36)(29,37)(32,40)(34,35);(3,7)(5,8)(9,12)(10,17)(11,22)(13,26)(14,29)(15,27)(16,30)(18,25)(19,34)(20,32)(21,35)(23,36)(24,37)(28,38)(31,40)(33,39);(3,7)(5,8)(11,23)(16,31)(21,33)(22,36)(24,28)(30,40)(35,39)(37,38);(2,6)(5,8)Polycyclic presentation:
2^9|a^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=dhi,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377255);