Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) order 8192 |
| 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 | C2×#41 | D4 |
| 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)(3,10)(4,6)(5,9)(7,11)(8,13)(12,15)(14,16)(17,18,21,26,22,24,20,25)(19,27,29,23,30,32,28,31);(2,6)(4,13)(7,14)(8,9)(10,15)(12,16)(17,19,22,30)(18,23,24,31)(20,29,21,28)(25,32,26,27);(1,3)(2,7)(4,12)(5,11)(6,14)(8,15)(9,10)(13,16)(18,24)(23,31)(25,26)(27,32);(3,11)(7,10)(12,16)(14,15)Polycyclic presentation:
2^8|a^2=f,b^2=gh,e^2=h,f^2=gh,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 | #221271, #221272, #221275, … |
| maximal normal subgroup of | #221271, #221272, #221275, … |
| maximal quotient of | #160637, #160639, #160640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10749);