Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | Aut(G) = #72261435 (order 2048) · Inn(G) = H(3,𝔽4) · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2782 |
| prime radical | #575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 89 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:
(1,2,6,10,7,12,5,11)(3,13,16,8,17,24,15,25)(4,18,22,9,20,28,21,27)(14,29,30,26,31,23,19,32);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,26)(10,24)(11,13)(12,25)(14,22)(18,29)(20,30)(21,31)(23,28)(27,32);(1,4,5,21,7,20,6,22)(2,9,10,28,12,27,11,18)(3,14,16,30,17,31,15,19)(8,23,13,26,25,29,24,32);(3,15,17,16)(4,20)(8,24,25,13)(9,27)(14,30,31,19)(18,28)(21,22)(23,26,29,32)Polycyclic presentation:
2^8|a^2=efg,c^2=e,d^2=ef,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,e^b=eh,g^b=gh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151252, #151253, C2×#32703, … |
| maximal normal subgroup of | #151252, #151253, C2×#32703, … |
| maximal quotient of | #384930, #386370, #387234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25691);