Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6444406 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1970 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)(3,7)(4,8)(5,6)(9,10,15,20,16,21,14,19)(11,22,24,12,25,27,17,28)(13,18,30,35,32,37,31,36)(23,38,29,26,39,34,33,40);(2,6)(7,8)(9,11)(10,12)(13,29)(14,24)(15,17)(16,25)(18,34)(19,27)(20,22)(21,28)(23,30)(26,35)(31,39)(32,33)(36,40)(37,38);(1,3,5,4)(2,7,6,8)(9,12)(10,17,21,24)(11,19,25,20)(13,26)(14,22)(15,27)(16,28)(18,33,37,29)(23,36,39,35)(30,34)(31,38)(32,40);(1,4,5,3)(2,8,6,7)(9,13,14,31,16,32,15,30)(10,18,19,36,21,37,20,35)(11,23,17,33,25,39,24,29)(12,26,22,38,28,40,27,34);(9,14,16,15)(10,19,21,20)(11,24,25,17)(12,27,28,22)(13,30,32,31)(18,35,37,36)(23,33,39,29)(26,38,40,34)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364910);