Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470415 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2743 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)(11,21,23,32)(12,26,28,34)(13,19,14,18)(16,25,31,24)(17,29,33,27)(22,36,37,30)(35,39,40,38);(1,3)(2,5)(4,7)(6,8)(9,11,13,24,15,23,14,25)(10,16,18,21,20,31,19,32)(12,22,27,38,28,37,29,39)(17,30,26,35,33,36,34,40);(3,7)(5,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,26)(19,34)(20,33)(21,35)(23,37)(24,38)(25,39)(31,36)(32,40);(3,7)(5,8);(2,6)(5,8)(11,23)(16,31)(21,32)(22,37)(24,25)(30,36)(35,40)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,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=dh,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377612);