Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#386 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
852 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)(3,7,12,21)(4,14,15,26)(5,10)(6,9)(8,16,23,18)(11,27)(13,22,17,25)(19,24)(20,30)(28,31)(29,32);(1,3)(2,7,9,21)(4,11)(5,17)(6,12)(8,20,23,32)(10,25,24,22)(13,19)(14,31,26,27)(15,28)(16,30)(18,29);(1,4)(2,8)(3,11)(5,18)(6,15)(7,20)(9,23)(10,26)(12,28)(13,30)(14,24)(16,19)(17,29)(21,32)(22,31)(25,27);(3,12)(7,21)(11,28)(13,17)(20,32)(22,25)(27,31)(29,30)Polycyclic presentation:
2^8|a^2=g,b^2=eg,f^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #359543, #359617, #359618, … |
| maximal normal subgroup of | #359543, #359617, #359618, … |
| maximal quotient of | #196173, #196174, #196175, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13988);