Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,12)(11,15,16,14)(17,18,21,20)(19,22,24,23);(1,3,6,4)(2,7,8,5)(9,10)(11,15)(12,13)(14,16)(18,20)(22,23);(2,8)(3,4)(9,11,13,16)(10,14,12,15)(17,18,21,20)(19,23,24,22);(1,4,6,3)(2,5,8,7)(9,11)(10,15)(12,14)(13,16)(17,19,21,24)(18,22,20,23);(1,5)(2,4)(3,8)(6,7)(9,12,13,10)(11,15,16,14)(17,20,21,18)(19,22,24,23)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=f,d^2=fg,e^2=f,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 | #130532, #254283, #254306, … |
| maximal normal subgroup of | #130532, #254283, #254306, … |
| maximal quotient of | #519182, #519194, #519210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29327);