Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) = #72252039 (order 2048) · Inn(G) = D42 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2358 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
498 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,12,21,10)(3,9,15,24,16,23,14,7)(4,17,18,27,20,26,19,8)(13,28,30,32,31,22,29,25);(1,3,5,15,6,16,21,14)(2,7,11,9,12,24,10,23)(4,13,18,30,20,31,19,29)(8,22,17,25,27,28,26,32);(1,4)(2,8)(3,13)(5,18)(6,20)(7,22)(9,25)(10,26)(11,17)(12,27)(14,29)(15,30)(16,31)(19,21)(23,32)(24,28);(1,3)(2,9)(4,13)(5,15)(6,16)(7,10)(8,25)(11,24)(12,23)(14,21)(17,28)(18,30)(19,29)(20,31)(22,26)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,f^c=fh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152461, #152464, #152553, … |
| maximal normal subgroup of | #152461, #152464, #152553, … |
| maximal quotient of | #226934, #226935, #226936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15886);