Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115392 |
| 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 | #60561 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,12,6)(4,8)(5,7)(10,15,11,14)(13,16)(17,18)(19,25,26,31)(20,23)(21,24)(22,27,30,28)(29,32);(1,3)(2,6)(4,10)(5,12)(7,9)(8,14)(11,13)(15,16)(22,30)(25,31);(3,10)(4,13)(6,14)(8,16)(9,15)(11,12)(17,19)(18,22)(20,28)(21,26)(23,31)(24,30)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fh,g^b=gh,d^c=di,e^c=ei,f^c=fh,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72226722, #72316961, #72338230, … |
| maximal normal subgroup of | #72226722, #72316961, #72338230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22588);