Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #148249 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11087 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
928 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)(3,7)(4,14,16,8)(5,10)(6,9)(11,29,30,21)(12,27)(13,22)(15,24,31,32)(17,28,19,25)(18,23)(20,26);(1,3)(2,7)(4,15,16,31)(5,18,20,12)(6,13)(8,24,14,32)(9,22)(10,27,26,23)(11,17)(19,30)(21,25)(28,29);(1,4)(2,8)(3,11)(5,19)(6,16)(7,21)(9,14)(10,28)(12,31)(13,30)(15,18)(17,20)(22,29)(23,32)(24,27)(25,26)Polycyclic presentation:
2^9|a^2=g,b^2=d,f^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72238712, #72238717, #72238723, … |
| maximal normal subgroup of | #72238712, #72238717, #72238723, … |
| maximal quotient of | #72251671 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55445);