Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7487864 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #52004 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,12)(4,8)(5,9)(6,11)(7,13)(10,15)(14,16);(1,3,6,14)(2,7,11,8)(4,15,13,5)(9,12,10,16);(7,8)(12,16);(1,4,5,3,6,13,15,14)(2,8,9,12,11,7,10,16);(2,9,11,10)(7,16,8,12)Polycyclic presentation:
2^9|b^2=i,d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ef,f^a=fi,h^a=hi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72215721, #72216600, #72216603, … |
| maximal normal subgroup of | #72215721, #72216600, #72216603, … |
| maximal quotient of | #72250784, C2×#7487865, #72256179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7395061);