Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479742 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17232 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,9,6,11)(3,13,14,25,16,8,17,23)(4,18,20,10,21,28,22,26)(15,30,29,24,19,32,31,27);(1,3,6,14,7,16,5,17)(2,8,11,23,9,13,12,25)(4,19,20,29,21,15,22,31)(10,27,28,32,26,24,18,30);(2,9)(3,14)(5,6)(8,23)(13,25)(15,29)(16,17)(18,28)(19,31)(20,22)(24,30)(27,32);(1,4)(2,10)(3,15)(5,20)(6,22)(7,21)(8,24)(9,26)(11,18)(12,28)(13,27)(14,29)(16,19)(17,31)(23,32)(25,30)Polycyclic presentation:
2^9|a^2=fg,b^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,d^c=de,g^c=gi,h^c=hi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251686, #72251691, #72251701, … |
| maximal normal subgroup of | #72251686, #72251691, #72251701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386938);