Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7339796 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #48088 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,6,8,12)(7,14,16,9);(1,2)(3,9)(4,12)(5,8)(6,13)(7,15)(10,14)(11,16);(3,10)(4,13)(6,12)(7,16);(1,3,4,11,5,10,13,15)(2,7,12,9,8,16,6,14);(1,2)(3,7)(4,12)(5,8)(6,13)(9,11)(10,16)(14,15)Polycyclic presentation:
2^9|a^2=g,d^2=fg,f^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,h^d=hi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338936, #72338939 |
| maximal normal subgroup of | #72338936, #72338939 |
| maximal quotient of | #72319417, #72338490, #72338522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7246992);