Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33469 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#442 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C4×#442 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,11,23,12)(3,13,16,30,18,31,17,25)(4,10,20,27,19,28,5,21)(8,22,24,14,26,32,29,15);(1,3)(2,8,11,26)(4,14)(5,22)(6,16)(7,18)(9,24,12,29)(10,25,28,30)(13,21,31,27)(15,19)(17,23)(20,32);(2,9,11,12)(4,19)(5,20)(8,24,26,29)(10,21,28,27)(13,25,31,30)(14,15)(22,32);(1,4,6,20,7,19,23,5)(2,10,9,27,11,28,12,21)(3,14,16,32,18,15,17,22)(8,25,24,13,26,30,29,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=ch,e^a=eh,g^a=gi,h^a=hi,e^b=ehi,h^b=hi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72214092, #72214585, … |
| maximal normal subgroup of | #72214039, #72214092, #72214585, … |
| maximal quotient of | #72216219, #72225334, C3×#493248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400444);