Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C22×C22⋊C4 · Out(G) = C23×D4 |
| 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 | |
438 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)(17,18,22,26,23,19,21,25)(20,27,28,32,30,24,29,31);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,23,18)(20,27,30,24)(21,26,22,25)(28,32,29,31);(2,7)(6,14)(8,15)(9,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31);(17,21,23,22)(18,25,19,26)(20,28,30,29)(24,31,27,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^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 | #221326, #221327, #221331, … |
| maximal normal subgroup of | #221326, #221327, #221331, … |
| maximal quotient of | #160633, #160639, #160646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10670);