Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(5,8)(9,10)(11,12)(13,15)(14,16)(17,18,21,20)(19,23,24,22);(1,2)(3,8)(4,6)(5,7)(9,11,14,13)(10,15,16,12);(1,2,4,6)(3,5,7,8)(9,11,14,13)(10,12,16,15)(17,19,21,24)(18,22,20,23);(1,2)(3,5)(4,6)(7,8)(9,12,14,15)(10,11,16,13)(17,20,21,18)(19,23,24,22);(1,3)(2,5)(4,7)(6,8)(9,13,14,11)(10,15,16,12)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#44100, #7668690, #7668714, … |
| maximal normal subgroup of | C2×#44100, #7668690, #7668714, … |
| maximal quotient of | #535523, #535734, #536221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37088);