Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | Aut(G) = #72340198 (order 4096) · Inn(G) = #2631 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2631 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
334 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 97 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:
(1,2,6,10,7,11,20,5)(3,12,14,24,16,23,15,8)(4,17,18,26,19,27,21,9)(13,25,29,28,31,22,30,32);(1,3)(2,8)(4,13)(5,12)(6,15)(7,16)(9,25)(10,23)(11,24)(14,20)(17,32)(18,30)(19,31)(21,29)(22,26)(27,28);(1,4,7,19)(2,9,11,26)(3,13,16,31)(5,17,10,27)(6,21,20,18)(8,22,24,25)(12,28,23,32)(14,30,15,29);(1,5,20,11,7,10,6,2)(3,12,14,24,16,23,15,8)(4,9,21,27,19,26,18,17)(13,25,29,28,31,22,30,32)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=fh,e^2=h,f^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,d^c=de,e^c=eh,f^c=fh,g^d=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #143163, #143165, #143167, … |
| maximal normal subgroup of | #143163, #143165, #143167, … |
| maximal quotient of | #331009, #331011, #331013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23046);