Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(13,14,18,17)(15,20,19,16);(5,6,10,8)(7,12,9,11)(13,15)(14,16)(17,20)(18,19);(1,2,3,4)(5,7,10,9)(6,11,8,12)(13,16,18,20)(14,19,17,15);(5,8,10,6)(7,11,9,12)(13,17,18,14)(15,20,19,16);(5,9,10,7)(6,12,8,11)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=gh,d^2=fh,e^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #98248, #98249, #130939, … |
| maximal normal subgroup of | #98248, #98249, #130939, … |
| maximal quotient of | #519089, #519272, C4⋊C4×C4⋊Q8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29548);