Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,19)(8,23)(9,16)(10,17)(11,14)(12,18)(13,22)(15,20)(21,35)(24,32)(25,31)(26,33)(27,30)(28,34)(29,36);(1,2,3,4)(5,7,10,22)(6,13,17,19)(8,12,20,11)(9,25,26,24)(14,15,18,23)(16,32,33,31)(21,28,36,27)(29,34,35,30);(6,14)(7,20)(8,22)(11,12)(15,23)(16,30)(17,18)(21,25)(24,36)(27,28)(31,32)(33,34);(5,8)(6,15)(7,12)(9,21)(10,20)(11,22)(13,18)(14,19)(16,29)(17,23)(24,27)(25,28)(26,36)(30,31)(32,34)(33,35);(5,9)(6,16)(7,21)(8,24)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(17,33)(18,34)(19,35)(20,25)(22,36)(23,32)Polycyclic presentation:
2^9|b^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481186);