Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| 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 | |
2340 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)(3,5)(4,6)(7,8)(9,10)(11,22)(12,28)(13,32)(14,21)(15,20)(16,19)(17,26)(18,30)(23,24)(25,37)(27,38)(29,39)(31,34)(33,40)(35,36);(1,3,5,2)(4,7,8,6)(9,11)(10,17)(12,29)(13,33)(14,25)(15,26)(16,27)(18,24)(19,37)(20,22)(21,38)(23,32)(28,35)(30,40)(31,36)(34,39);(1,4,5,8)(2,6,3,7)(9,12,15,31)(10,18,20,13)(11,23,26,40)(14,30,16,32)(17,35,22,39)(19,28,21,34)(24,37,33,38)(25,29,27,36);(2,3)(6,7)(9,13)(10,12)(11,24)(14,34)(15,18)(16,28)(17,36)(19,30)(20,31)(21,32)(22,29)(23,37)(25,35)(26,33)(27,39)(38,40);(9,14,15,16)(10,19,20,21)(11,25,26,27)(12,30,31,32)(13,34,18,28)(17,37,22,38)(23,29,40,36)(24,35,33,39)Polycyclic presentation:
2^9|b^2=fi,c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,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, 7409020);