Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152116 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12697 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #60798 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #3444 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
826 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 826 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,4,9)(3,7,11,14)(5,8,12,15)(6,10,13,16);(2,7)(3,5)(4,12)(8,10)(11,13)(14,15);(2,8,10,7)(4,13)(9,14,16,15)(11,12)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,f^c=fi,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72240970, #72240971, #72240974, … |
| maximal normal subgroup of | #72240970, #72240971, #72240974, … |
| maximal quotient of | #72279237, C2×#152116, #72293066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59312);