Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #392 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
974 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,19)(14,22)(15,23)(17,27)(20,26)(21,25)(24,30)(28,29)(31,32);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,24)(10,13)(11,27)(12,23)(18,28)(19,29)(20,31)(21,30)(22,26)(25,32);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,18)(12,26)(13,28)(15,29)(16,31)(17,30)(23,24)(27,32);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,23,27)(9,25,26,18)(14,29,30,31)(22,28,24,32)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #359071, #375827, #375838, … |
| maximal normal subgroup of | #359071, #375827, #375838, … |
| maximal quotient of | #232508, #232512, #232513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16400);