Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #113020 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12,9,15,30,3)(4,19,22,29,10,18,8,14)(5,11,25,23,6,26,28,16)(13,20,31,27,32,21,17,24);(1,3)(2,10)(4,15)(5,23)(6,16)(7,29)(8,18)(9,12)(11,20)(13,25)(14,30)(17,27)(19,22)(21,26)(24,31)(28,32);(2,11)(3,16)(5,24)(6,27)(12,23)(13,14)(15,26)(17,18)(19,31)(20,25)(21,28)(29,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,e^c=eh,f^c=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72213800, #72213848, #72215564, … |
| maximal normal subgroup of | #72213800, #72213848, #72215564, … |
| maximal quotient of | C2×#113024, #72240263 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20220);