Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) = C26×#120 (order 2048) · Inn(G) = C22×C22⋊C4 · Out(G) = C25 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | #3515 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
439 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,22,25,23,19,21,27)(20,26,30,32,28,24,29,31);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,23,18)(20,26,28,24)(21,25,22,27)(29,31,30,32);(2,7)(6,14)(8,15)(9,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,28)(25,31)(27,32);(20,28)(24,26)(29,30)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,f^2=g,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh,d^c=dg49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17680, #6343814, #6343816, … |
| maximal normal subgroup of | C2×#17680, #6343814, #6343816, … |
| maximal quotient of | #160631, #160637, #160643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10668);