Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30236 |
| automorphism | Aut(G) = C2×#72237738 (order 6144) · Inn(G) = #2644 · Out(G) = C22×A4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2644 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,12,5,10)(3,13,14,23,16,21,17,8)(4,9,20,25,19,26,18,24)(15,27,28,32,29,31,30,22);(1,3,6,16)(2,8)(4,15,19,29)(5,14,7,17)(9,22)(10,13)(11,21)(12,23)(18,28,20,30)(24,27)(25,31)(26,32);(3,14)(4,18)(5,7)(8,21)(9,24)(10,11)(13,23)(15,29)(16,17)(19,20)(25,26)(27,31);(1,4,5,18,6,19,7,20)(2,9,10,24,12,26,11,25)(3,15,14,28,16,29,17,30)(8,22,13,27,23,32,21,31)Polycyclic presentation:
2^8|a^2=fg,b^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,g^b=gh,d^c=de,e^c=eh,g^c=gh34 conjugacy classes — view the full character table.
| Fitting subgroup of | #30236⋊C3 |
| maximal subgroup of | C2×#30236, #6435889, #6435890, … |
| maximal normal subgroup of | C2×#30236, #6435889, #6435890, … |
| maximal quotient of | #380189, #380394, #382343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23224);