Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| 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 | |
1947 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,25)(13,31)(14,19)(15,18)(16,24)(17,30)(21,35)(22,34)(23,32)(26,37)(27,28)(29,36)(33,38)(39,40);(2,5)(3,7)(9,11)(10,16)(12,26)(13,22)(14,24)(15,32)(17,35)(18,34)(19,20)(21,27)(23,31)(25,33)(28,40)(29,38)(30,39)(36,37);(9,12)(10,17)(11,21)(13,27)(14,29)(15,30)(16,33)(18,36)(19,28)(20,37)(22,38)(23,40)(24,39)(25,31)(26,32)(34,35);(2,5)(6,8)(9,13,14,15)(10,18,19,31)(11,22,24,32)(12,27,29,30)(16,34,20,23)(17,36,28,25)(21,38,39,26)(33,35,37,40);(1,3,4,7)(2,6,5,8)(9,10)(11,23)(12,28)(13,18)(14,19)(15,31)(16,22)(17,29)(20,32)(21,35)(24,34)(25,27)(26,33)(30,36)(37,38)(39,40)Polycyclic presentation:
2^9|d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,e^c=ei,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580145);