Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #58990 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3031 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)(3,15)(4,18)(5,22)(6,9)(7,13)(8,14)(10,16)(11,20)(12,23)(17,26)(19,27)(21,25)(24,28)(29,31)(30,32);(2,9)(3,16)(6,7)(11,18)(12,22)(13,14)(15,23)(17,31)(20,24)(25,28)(27,30)(29,32);(1,3)(2,10)(4,17)(5,14)(6,16)(7,22)(8,12)(9,23)(11,26)(13,15)(18,29)(19,28)(20,30)(21,27)(24,31)(25,32);(1,4,8,21)(2,11,14,28)(3,17,12,27)(5,19,10,26)(6,20,7,24)(9,25,13,18)(15,29,23,32)(16,30,22,31);(1,5)(2,12)(3,14)(4,19)(6,15)(7,23)(8,10)(9,22)(11,27)(13,16)(17,28)(18,30)(20,29)(21,26)(24,32)(25,31)Polycyclic presentation:
2^9|d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,g^c=gi,h^c=hi,g^d=gi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421338);