Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350061 |
| 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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2934 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)(5,14)(6,10)(7,11)(12,15)(13,16)(19,26)(20,21)(23,30)(24,28)(27,29)(31,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(17,18,22,25)(19,23,27,32)(20,24,21,28)(26,31,29,30);(17,19)(18,23)(20,26)(21,29)(22,27)(24,31)(25,32)(28,30);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,31,32,30);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72339736, #72339782, #72339796, … |
| maximal normal subgroup of | #72339736, #72339782, #72339796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257399);