Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | |
2628 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)(3,7)(4,6)(5,8)(9,10)(11,22)(12,27)(13,21)(14,19)(15,20)(16,25)(17,31)(18,32)(23,40)(24,35)(26,37)(28,34)(29,38)(30,36)(33,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,28)(13,24)(14,29)(15,25)(17,19)(18,35)(20,22)(21,37)(23,31)(26,32)(27,39)(30,33)(34,36)(38,40);(9,12)(10,17)(11,23)(13,29)(14,25)(15,30)(16,34)(18,36)(19,37)(20,38)(21,27)(22,39)(24,28)(26,33)(31,32)(35,40);(2,6)(5,7)(9,13,15,32)(10,18,20,21)(11,24,25,26)(12,29,30,31)(14,33,23,28)(16,35,22,37)(17,36,38,27)(19,34,40,39);(3,8)(5,7)(9,14,15,23)(10,19,20,40)(11,12,25,30)(13,28,32,33)(16,36,22,27)(17,37,38,35)(18,39,21,34)(24,31,26,29)Polycyclic presentation:
2^9|d^2=i,e^2=i,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,g^c=gi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575163);