Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2666 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)(4,6)(5,8)(9,10,16,20,14,12,15,19)(11,21,32,22,24,17,25,36)(13,27,31,26,29,35,30,18)(23,38,40,33,28,37,39,34);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,21)(13,28)(14,24)(15,25)(16,32)(18,34)(19,22)(20,36)(23,29)(26,33)(27,38)(30,40)(31,39)(35,37);(9,12,14,10)(11,22,24,36)(13,26,29,18)(15,19,16,20)(17,32,21,25)(23,37,28,38)(27,30,35,31)(33,39,34,40);(2,6)(3,8)(9,13)(10,18)(11,23)(12,26)(14,29)(15,31)(16,30)(17,33)(19,27)(20,35)(21,34)(22,37)(24,28)(25,40)(32,39)(36,38);(11,24)(17,21)(22,36)(23,28)(25,32)(33,34)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=gi,c^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410513);