Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #248551 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,7)(3,5)(4,8)(9,10)(11,13)(12,14)(15,16);(1,2)(3,7)(4,6)(5,8)(9,11)(10,13)(12,15)(14,16);(1,3,6,5)(2,7,4,8)(13,16);(1,4,6,2)(3,8,5,7)Polycyclic presentation:
2^9|c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,e^d=eh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72257733, #72263885, #72263886, … |
| maximal normal subgroup of | #72257733, #72263885, #72263886, … |
| maximal quotient of | #72264002, #72264255 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 155756);