Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347252 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,10)(3,11,12,24)(4,15,18,9)(6,8,19,23)(7,16,21,13)(14,25,26,32)(17,28,30,22)(20,29,31,27);(1,3)(2,7)(4,16)(5,12)(6,14)(8,20)(9,24)(10,21)(11,15)(13,18)(17,29)(19,26)(22,32)(23,31)(25,28)(27,30);(4,17)(9,22)(11,25)(13,27)(15,28)(16,29)(18,30)(24,32);(2,8)(7,20)(9,22)(10,23)(11,25)(15,28)(21,31)(24,32);(3,12)(7,21)(11,24)(13,16)(14,26)(20,31)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=h,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232826, #72233045, #72234673, … |
| maximal normal subgroup of | #72232826, #72233045, #72234673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254449);