Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1855 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)(4,8)(5,6)(9,10)(11,22)(12,25)(13,18)(14,19)(15,21)(16,17)(20,33)(23,38)(24,27)(26,39)(28,40)(29,32)(30,36)(31,37)(34,35);(2,6)(7,8)(9,11)(10,16)(12,21)(13,28)(14,17)(15,24)(18,26)(19,22)(20,27)(23,32)(25,33)(29,34)(30,39)(31,35)(36,40)(37,38);(9,12,14,27)(10,17,19,11)(13,26,30,40)(15,22,33,16)(18,35,36,23)(20,24,21,25)(28,37,39,29)(31,38,32,34);(1,3,5,4)(2,7,6,8)(9,10)(11,16)(12,25)(13,29)(14,19)(15,20)(17,22)(18,32)(21,33)(23,26)(24,27)(28,34)(30,37)(31,36)(35,40)(38,39);(1,4,5,3)(2,8,6,7)(9,13,14,30)(10,18,19,36)(11,23,17,35)(12,26,27,40)(15,31,33,32)(16,34,22,38)(20,29,21,37)(24,28,25,39)Polycyclic presentation:
2^9|c^2=i,d^2=fi,e^2=f,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,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485107);