Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153149 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13346 |
| commutator, Frattini, cosocle | #532 | C23 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 599 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:
(2,7)(3,5)(4,11)(9,10)(12,14)(15,16);(2,8)(4,12)(7,15)(9,16)(10,13)(11,14);(1,2)(3,9)(4,13)(5,7)(6,10)(8,12)(11,15)(14,16)Polycyclic presentation:
2^9|d^2=h,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fgh,g^a=gi,c^b=cf,d^b=dh,f^b=fi,g^b=gh,d^c=dg,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72209905, #72240970, #72283303, … |
| maximal normal subgroup of | #72209905, #72240970, #72283303, … |
| maximal quotient of | #72213442, #72227849, #72228012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60345);