Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35346 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,9)(5,11)(6,19)(7,12)(8,13)(14,25)(15,23)(16,22)(17,24)(18,28)(20,27)(21,29)(26,30)(31,32);(1,3)(2,8)(4,13)(5,14)(6,16)(7,15)(9,23)(10,19)(11,26)(12,22)(17,29)(18,31)(20,30)(21,28)(24,25)(27,32);(1,4,7,6)(2,9,12,19)(3,13,15,16)(5,17,18,20)(8,10,22,23)(11,24,28,27)(14,29,31,30)(21,25,26,32);(1,3)(2,10)(4,16)(5,14)(6,13)(7,15)(8,19)(9,22)(11,25)(12,23)(17,30)(18,31)(20,29)(21,27)(24,26)(28,32);(1,5,7,18)(2,11,12,28)(3,14,15,31)(4,17,6,20)(8,21,22,26)(9,24,19,27)(10,25,23,32)(13,29,16,30)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dhi,g^d=gi,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233777, #72261130, #72263053, … |
| maximal normal subgroup of | #72233777, #72261130, #72263053, … |
| maximal quotient of | #72263057, #72318059, #72318131, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407061);