Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349016 |
| 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 | |
3209 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)(17,18)(19,27)(20,26)(21,24)(22,25)(23,29)(28,31)(30,32);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(17,19)(18,23)(20,30)(21,29)(22,28)(24,31)(25,32)(26,27);(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(3,10)(6,15)(9,16)(11,13)(18,25)(19,28)(24,26)(29,30);(2,7)(6,15)(8,12)(9,16)(18,25)(23,32)(24,26)(27,31)Polycyclic presentation:
2^9|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=dh,f^b=fh,g^b=gi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317645, #72318515, #72318616, … |
| maximal normal subgroup of | #72317645, #72318515, #72318616, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256212);