Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30871 |
| automorphism | Aut(G) = #72340866 (order 12288) · Inn(G) = #2675 · Out(G) = C22×S4 |
| 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 | |
423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,14)(4,18)(5,11)(6,24)(7,12)(8,13)(9,16)(10,22)(15,28)(17,26)(19,31)(20,27)(21,30)(23,29)(25,32);(1,3)(2,9)(4,19)(5,6)(7,16)(8,17)(10,27)(11,13)(12,26)(14,24)(15,21)(18,25)(20,23)(22,32)(28,29)(30,31);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22)(8,23)(9,25)(11,28)(12,29)(13,30)(14,31)(16,19)(17,32)(18,24)(26,27);(1,5,7,17,8,16,6,3)(2,11,24,14,13,9,12,26)(4,20,22,32,23,19,21,15)(10,28,18,31,30,25,29,27)Polycyclic presentation:
2^8|d^2=eh,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 | #30872⋊C3 |
| maximal subgroup of | #151264, #151266, #151268, … |
| maximal normal subgroup of | #151264, #151266, #151268, … |
| maximal quotient of | #339704, #339705, #339711, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23860);