Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479799 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17255 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,8,16,31)(4,17)(5,10)(7,12,19,29)(9,27)(13,28)(14,21)(15,30,26,20)(18,25)(22,23)(24,32);(1,3,7,26,6,16,19,15)(2,8,29,30,11,31,12,20)(4,18,21,10,22,13,27,32)(5,23,25,14,24,17,28,9);(1,4,19,21,6,22,7,27)(2,9,29,17,11,14,12,23)(3,13,15,32,16,18,26,10)(5,20,28,8,24,30,25,31);(1,5,6,24)(2,10,11,32)(3,14,16,9)(4,20,22,30)(7,28,19,25)(8,21,31,27)(12,13,29,18)(15,17,26,23)Polycyclic presentation:
2^9|a^2=eg,b^2=efh,c^2=ef,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251686, #72251695, #72251696, … |
| maximal normal subgroup of | #72251686, #72251695, #72251696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386996);