Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367790 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1670 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)(3,8)(4,9)(5,7)(6,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,31)(27,29)(30,32);(2,7)(3,9)(4,13)(5,14)(8,15)(10,11)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(1,3,6,9)(2,8,10,4)(5,12,14,16)(7,13,11,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,6,8)(2,9,10,3)(5,15,14,13)(7,16,11,12);(2,10)(3,9)(7,11)(12,16)(18,24)(19,26)(23,25)(27,32)Polycyclic presentation:
2^9|c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72284241, #72338364, #72338422, … |
| maximal normal subgroup of | #72284241, #72338364, #72338422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274998);