Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C25 |
| commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,28,31)(20,25,21,24)(27,32,29,30);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,30)(25,32)(26,31);(3,10)(6,9)(11,16)(13,15)(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32);(2,7)(6,9)(8,14)(13,15)(19,28)(23,31)(27,29)(30,32);(17,21,22,20)(18,25,26,24)(19,29,28,27)(23,32,31,30);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)Polycyclic presentation:
2^8|a^2=h,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=cg,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #186526, #186530, #186544, … |
| maximal normal subgroup of | #186526, #186530, #186544, … |
| maximal quotient of | #7625991, #7626000, #7626047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55749);