Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #114446 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1129 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,13,8,3,6,16)(4,17,20,28,22,29,31,30)(7,11,23,10,26,15,25,12)(9,19,27,32,14,24,18,21);(1,3)(2,8)(4,18)(5,10)(6,12)(7,15)(9,24)(11,26)(13,23)(14,19)(16,25)(17,31)(20,29)(21,30)(22,27)(28,32);(1,4)(2,9)(3,14)(5,19)(6,24)(7,21)(8,22)(10,27)(11,17)(12,18)(13,28)(15,29)(16,30)(20,23)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,f^c=fi,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72267801 |
| maximal normal subgroup of | #72267801 |
| maximal quotient of | #72267823 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 21645);