Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437445 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3938 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)(3,8)(4,6)(5,7)(9,10)(11,22)(12,18)(13,19)(14,20)(15,30)(16,21)(17,26)(23,32)(24,34)(25,33)(27,38)(28,35)(29,36)(31,40)(37,39);(9,11)(10,17)(12,23)(13,24)(14,26)(15,25)(16,27)(18,34)(19,37)(20,38)(21,33)(22,30)(28,40)(29,39)(31,36)(32,35);(1,3,5,4)(2,6,7,8)(9,12,16,29)(10,18,21,36)(11,23,27,39)(13,14,28,15)(17,31,33,34)(19,20,35,30)(22,32,38,37)(24,26,40,25);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,24)(12,14)(15,29)(16,28)(17,32)(18,20)(21,35)(22,34)(23,26)(25,39)(27,40)(30,36)(31,38)(33,37);(9,14,16,15)(10,20,21,30)(11,25,27,26)(12,28,29,13)(17,22,33,38)(18,35,36,19)(23,24,39,40)(31,32,34,37)Polycyclic presentation:
2^9|c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318196, #72318324, #72318405, … |
| maximal normal subgroup of | #72318196, #72318324, #72318405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358946);