Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| 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 | #3512 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(18,22,24,27)(23,29,30,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,18)(19,23)(20,27)(21,24)(22,28)(25,30)(26,32)(29,31);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(19,25)(23,30)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=g,e^2=h,f^2=g,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #220603, #220604, #220610, … |
| maximal normal subgroup of | #220603, #220604, #220610, … |
| maximal quotient of | #163727, #163735, #163743, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10675);