Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486694 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #18203 |
| commutator, Frattini, cosocle | C2×M16 | C24 |
| prime radical | #62283 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,12,5,21)(3,13,22,25,15,27,14,9)(4,17,18,26,19,8,20,10)(16,31,32,23,30,24,29,28);(2,8)(3,14)(4,18)(5,7)(9,24)(10,21)(11,26)(12,17)(13,23)(15,22)(19,20)(25,31)(27,28)(29,32);(3,15)(4,19)(9,25)(13,27)(14,22)(18,20)(23,28)(24,31);(1,3,5,14,6,15,7,22)(2,9,11,13,12,25,21,27)(4,16,20,29,19,30,18,32)(8,23,10,24,17,28,26,31)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=g,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,g^a=gi,h^a=hi,d^b=dg,e^b=egi,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231596, #72231700, #72231715, … |
| maximal normal subgroup of | #72231596, #72231700, #72231715, … |
| maximal quotient of | #72250607, #72250955, #72257001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393890);