Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #379 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#379 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1822 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)(3,12)(4,16,19,9)(5,10)(6,20)(7,11)(8,14)(13,27,29,26)(15,23)(17,31,18,25)(21,24)(22,28,30,32);(1,3)(2,8)(4,13)(5,14)(6,21)(7,15)(9,22)(10,23)(11,24)(12,20)(16,30)(17,28)(18,32)(19,29)(25,27)(26,31);(1,4)(2,9)(3,13)(5,17)(6,18)(7,19)(8,22)(10,25)(11,16)(12,26)(14,28)(15,29)(20,31)(21,32)(23,27)(24,30);(1,5,7,6)(2,10,11,20)(3,14,15,21)(4,17,19,18)(8,23,24,12)(9,25,16,31)(13,28,29,32)(22,27,30,26)Polycyclic presentation:
2^8|a^2=g,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146486, #146489, #146589, … |
| maximal normal subgroup of | #146486, #146489, #146589, … |
| maximal quotient of | #184970, #184971, #184972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13611);