Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6346727 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(5,8)(10,14)(11,12)(15,16);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,16)(13,15);(1,3)(2,5)(4,7)(6,8)(11,12)(14,16);(9,11,13,12)(10,14,15,16);(10,15)(14,16)Polycyclic presentation:
2^9|d^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232937, #72233388, #72233410, … |
| maximal normal subgroup of | #72232937, #72233388, #72233410, … |
| maximal quotient of | #72234733, #72260872, #72260979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253924);