Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574086 |
| 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 | |
1780 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)(11,16)(12,17)(13,19)(14,20)(15,22)(18,31)(21,34)(23,33)(24,35)(25,29)(26,40)(27,32)(28,38)(30,37)(36,39);(2,6)(7,8)(9,10)(11,16)(12,25)(13,18)(14,20)(15,23)(17,29)(19,31)(21,32)(22,33)(24,35)(26,36)(27,34)(28,37)(30,38)(39,40);(1,3,5,4)(2,7,6,8)(9,11,13,22,14,24,31,23)(10,15,18,16,20,33,19,35)(12,26,28,39,30,21,29,27)(17,36,25,40,37,32,38,34);(1,4,5,3)(2,8,6,7)(9,11)(10,16)(12,27)(13,22)(14,24)(15,19)(17,32)(18,33)(20,35)(21,29)(23,31)(25,34)(26,28)(30,39)(36,37)(38,40);(9,12,14,30)(10,17,20,37)(11,21,24,26)(13,28,31,29)(15,32,33,36)(16,34,35,40)(18,25,19,38)(22,27,23,39)Polycyclic presentation:
2^9|c^2=fg,d^2=f,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,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, 6492007);