Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62832 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2834 conjugacy classes of subgroups; only the 326 normal subgroups are shown.
This group has 2834 conjugacy classes of subgroups — too many to draw. Only its 326 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) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,5,6,7)(10,14)(11,12)(15,16);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15);(3,8)(5,7)(11,12)(14,16);(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232846, #72233149, #72233151, … |
| maximal normal subgroup of | #72232846, #72233149, #72233151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94025);