Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30871 |
| automorphism | Aut(G) = C2×#72237738 (order 6144) · Inn(G) = #2675 · Out(G) = C22×A4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2675 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(1,2,7,13)(3,14,19,28)(4,5,10,25)(6,12,26,11)(8,18,29,17)(9,21,24,22)(15,23,31,27)(16,20,32,30);(1,3)(2,8)(4,20)(5,23)(6,18)(7,19)(9,16)(10,30)(11,28)(12,14)(13,29)(15,22)(17,26)(21,31)(24,32)(25,27);(1,4)(2,9)(3,15)(5,12)(6,21)(7,10)(8,27)(11,25)(13,24)(14,16)(17,30)(18,20)(19,31)(22,26)(23,29)(28,32);(1,5,6,9,7,25,26,24)(2,10,11,22,13,4,12,21)(3,16,17,23,19,32,18,27)(8,15,14,30,29,31,28,20)Polycyclic presentation:
2^8|a^2=h,d^2=e,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,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #30873⋊C3 |
| maximal subgroup of | C2×#30873, #7502164, #7502165, … |
| maximal normal subgroup of | C2×#30873, #7502164, #7502165, … |
| maximal quotient of | #339707, #339708, #339709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23861);