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 | |
2591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,24)(12,27)(13,20)(14,21)(15,22)(16,34)(17,23)(18,26)(19,30)(25,38)(28,40)(29,39)(31,37)(32,35)(33,36);(1,3,5,4)(2,6,7,8)(9,11,15,26)(10,18,22,24)(12,28,25,32)(13,17,33,16)(14,30,29,31)(19,21,37,39)(20,34,36,23)(27,35,38,40);(9,12,15,25)(10,19,22,37)(11,14,26,29)(13,32,33,28)(16,31,17,30)(18,35,24,40)(20,21,36,39)(23,38,34,27);(1,4,5,3)(2,6,7,8)(9,13)(10,20)(11,17)(12,29)(14,25)(15,33)(16,26)(18,34)(19,35)(21,38)(22,36)(23,24)(27,39)(28,31)(30,32)(37,40);(9,14,15,29)(10,21,22,39)(11,25,26,12)(13,31,33,30)(16,28,17,32)(18,27,24,38)(19,20,37,36)(23,35,34,40)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,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, 6483943);