Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| 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:
(1,2,6,11,7,12,5,10)(3,13,16,24,17,23,15,8)(4,18,19,9,21,25,20,26)(14,22,28,31,30,32,29,27);(1,3)(2,8,12,24)(4,14)(5,15)(6,16)(7,17)(9,22,26,32)(10,23,11,13)(18,27,25,31)(19,28)(20,29)(21,30);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,18)(12,26)(13,27)(15,28)(16,29)(17,30)(23,31)(24,32);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,23,24,13)(9,25,26,18)(14,28,30,29)(22,31,32,27)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^b=fh,e^c=eh,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #354322, #355823, #356645, … |
| maximal normal subgroup of | #354322, #355823, #356645, … |
| maximal quotient of | #196078, #196079, #196080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13986);