Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347491 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(18,23)(20,21)(24,31)(25,26)(27,29)(30,32);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,32)(29,31);(17,19)(18,24)(20,27)(21,29)(22,28)(23,30)(25,31)(26,32);(2,7)(6,15)(8,12)(9,16);(2,7)(3,10)(8,12)(11,13)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,31,32,30)Polycyclic presentation:
2^9|e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,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 | #72284103, #72284113, #72284305, … |
| maximal normal subgroup of | #72284103, #72284113, #72284305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254714);