Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574107 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,9)(5,21)(6,10)(7,11)(8,15)(13,29)(14,25)(16,24)(17,23)(18,22)(19,27)(20,28)(26,31)(30,32);(1,2,6,21,7,11,5,10)(3,12,15,25,16,24,14,8)(4,9,22,27,20,28,19,18)(13,29,17,32,31,26,30,23);(1,3)(2,8)(4,17)(5,14)(6,15)(7,16)(9,26)(10,24)(11,25)(12,21)(13,19)(18,32)(20,30)(22,31)(23,27)(28,29);(1,4)(2,9)(3,13)(5,19)(6,22)(7,20)(8,23)(10,18)(11,28)(12,29)(14,30)(15,17)(16,31)(21,27)(24,26)(25,32);(1,2)(3,8)(4,18)(5,10)(6,21)(7,11)(9,22)(12,15)(13,29)(14,24)(16,25)(17,32)(19,28)(20,27)(23,30)(26,31)Polycyclic presentation:
2^9|b^2=fi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6481304);