Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153125 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2199 |
| commutator, Frattini, cosocle | #532 | C23 |
| socle | C2 | #13343 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 614 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,13)(4,9)(5,14)(6,15)(7,11)(8,12)(10,16);(2,9)(3,14)(4,16)(6,13)(10,12)(11,15);(1,3)(2,4)(5,10)(6,11)(7,9)(8,15)(12,16)(13,14)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72209817, #72209905, #72209931, … |
| maximal normal subgroup of | #72209905, #72337673, #72340623 |
| maximal quotient of | #72209813, #72213434, #72227652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60321);