Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574063 |
| 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 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,15)(9,14)(10,18)(11,17)(12,20)(16,22)(19,31)(21,30)(23,34)(24,36)(25,29)(26,35)(27,33)(28,32);(5,7,10,20,11,8,22,21)(6,12,16,13,17,30,18,15)(9,19,26,25,27,23,28,24)(14,29,32,31,33,36,35,34);(2,4)(5,8)(6,13)(7,11)(9,24)(10,21)(12,18)(14,29)(15,17)(16,30)(19,26)(20,22)(23,28)(25,27)(31,32)(33,36)(34,35);(5,9)(6,14)(7,19)(8,23)(10,26)(11,27)(12,29)(13,31)(15,34)(16,32)(17,33)(18,35)(20,25)(21,24)(22,28)(30,36);(5,8)(6,15)(7,11)(9,25)(10,21)(12,16)(13,17)(14,29)(18,30)(19,28)(20,22)(23,26)(24,27)(31,32)(33,36)(34,35)Polycyclic presentation:
2^9|b^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6481259);