Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365694 |
| 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 | |
1600 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)(4,7)(5,11)(6,8)(9,16)(12,15)(13,14)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,29)(31,32);(1,3)(2,4)(5,14)(6,12)(7,16)(8,13)(9,15)(10,11)(17,19)(18,22)(20,27)(21,26)(23,29)(24,30)(25,31)(28,32);(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,4,6,13)(2,7,8,14)(3,11,12,9)(5,15,16,10)(18,24)(22,30)(23,25)(29,31);(2,8)(3,12)(7,14)(9,11)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^9|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,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72339869, #72340114, #72340121, … |
| maximal normal subgroup of | #72339869, #72340114, #72340121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272898);