Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #113655 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,13,16,14)(12,15)(17,18)(19,27,30,32)(20,23)(21,25)(22,26)(24,29,31,28);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,16)(13,15)(18,23)(20,21)(24,27)(25,26)(28,29)(31,32);(1,4,5,12)(2,7,8,15)(3,10,11,16)(6,13,9,14)(17,19,22,30)(18,24,26,31)(20,29,21,28)(23,27,25,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=di,e^c=ei,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312121, #72312129, #72312800, … |
| maximal normal subgroup of | #72312121, #72312129, #72312800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20852);