Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | C22⋊#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7,12,23)(4,14)(5,17)(6,9)(8,16)(10,20)(11,27,29,31)(13,25,18,22)(15,26)(19,24)(21,30,32,28);(1,3)(2,7,9,23)(4,11)(5,18)(6,12)(8,21,24,32)(10,25,17,22)(13,20)(14,31,26,27)(15,29)(16,30)(19,28);(1,4)(2,8)(3,11)(5,19)(6,15)(7,21)(9,24)(10,26)(12,29)(13,30)(14,17)(16,20)(18,28)(22,31)(23,32)(25,27)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10792, #353608, #356478, … |
| maximal normal subgroup of | C2×#10792, #353608, #356478, … |
| maximal quotient of | #105680, #105681, #105684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3780);