Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477922 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17126 |
| 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 | |
875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,11,7,3)(4,13,16,9,17,27,22,24)(5,18,19,26,20,14,21,10)(15,25,30,28,31,23,32,29);(1,3,6,11,8,12,7,2)(4,13,16,24,17,27,22,9)(5,10,19,14,20,26,21,18)(15,23,30,28,31,25,32,29);(1,4,6,16,8,17,7,22)(2,9,3,13,11,24,12,27)(5,15,19,30,20,31,21,32)(10,25,14,29,26,23,18,28);(1,5)(2,10,11,26)(3,14,12,18)(4,15)(6,19)(7,21)(8,20)(9,23,24,25)(13,28,27,29)(16,30)(17,31)(22,32)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251783, #72251789, #72251795, … |
| maximal normal subgroup of | #72251783, #72251789, #72251795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385118);