Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112929 |
| 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 | |
1053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,11,22)(3,13)(4,17)(8,24,26,21)(9,14,12,23)(10,27,25,20)(15,30)(16,29)(18,32)(28,31);(1,2)(3,14)(4,8)(5,12)(6,9)(7,16)(10,18)(11,19)(13,22)(15,24)(17,25)(20,31)(21,30)(23,29)(26,32)(27,28);(2,8)(4,18)(7,20)(9,25)(10,12)(11,26)(14,24)(15,28)(17,32)(21,23)(22,27)(30,31)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,f^c=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213800, #72213902, #72215564, … |
| maximal normal subgroup of | #72213800, #72213902, #72215564, … |
| maximal quotient of | C2×#112929, #72240273 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20125);