Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6348967 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,25)(20,28)(21,23)(22,27)(24,29)(26,32)(30,31);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(20,29)(24,28)(25,32)(27,30);(17,19,21,26)(18,22,23,31)(20,30,29,27)(24,32,28,25);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,23)(22,31)(24,28)(25,32);(2,9)(7,16)(8,13)(10,15)Polycyclic presentation:
2^9|c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72306661, #72306713, #72306715, … |
| maximal normal subgroup of | #72306661, #72306713, #72306715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256181);