Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
2601 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)(11,15)(12,17)(13,16)(14,19)(18,29)(20,33)(21,31)(22,32)(23,30)(24,35)(25,36)(26,28)(27,34)(37,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,24)(13,20)(14,23)(16,33)(17,35)(18,32)(19,30)(21,28)(22,29)(25,37)(26,31)(27,38)(34,39)(36,40);(10,16)(11,20)(12,25)(13,29)(15,30)(17,26)(18,19)(21,37)(22,23)(24,38)(34,36)(35,40);(1,4,5,3)(2,6,7,8)(9,10)(11,15)(12,26)(13,18)(14,19)(16,29)(17,28)(20,32)(21,35)(22,33)(23,30)(24,31)(25,34)(27,36)(37,39)(38,40);(9,12,14,25)(10,17,19,36)(11,21,23,38)(13,27,29,28)(15,31,30,39)(16,34,18,26)(20,24,22,37)(32,40,33,35)Polycyclic presentation:
2^9|b^2=f,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483939);