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 | #2430 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,6,4)(9,10,13,12)(11,14,16,15)(18,19)(22,23);(1,3,5,6)(2,4,7,8)(10,12)(14,15)(17,18,21,19)(20,23,24,22);(1,4,5,8)(2,6,7,3)(9,11,13,16)(10,14,12,15)(17,19,21,18)(20,23,24,22);(1,4)(2,3)(5,8)(6,7)(9,12,13,10)(11,15,16,14)(17,20)(18,22)(19,23)(21,24);(1,3)(2,4)(5,6)(7,8)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=f,d^2=fg,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×#43690, #7667748, #7667899, … |
| maximal normal subgroup of | C2×#43690, #7667748, #7667899, … |
| maximal quotient of | #535495, #535602, #536168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 36678);