Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437969 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35528 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,17)(12,20)(13,19)(14,16)(15,22)(18,24)(21,23);(2,5)(6,7)(9,11,17,10,14,19,13,16)(12,21,24,15,22,18,23,20);(1,3,4,8)(2,6,5,7)(9,12,14,22)(10,15,16,20)(11,18,19,21)(13,24,17,23);(11,19)(13,17)(18,21)(23,24);(12,22)(15,20)(18,21)(23,24)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72251974, #72251976, #72252040, … |
| maximal normal subgroup of | #72251974, #72251976, #72252040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354940);