Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | #2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,13,14)(11,15,16,12)(17,18)(19,21)(20,23)(22,24);(2,6)(4,7)(11,16)(12,15)(18,19)(22,23);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,16)(14,15)(17,19,21,18)(20,23,24,22);(1,3,5,8)(2,7,6,4)(17,20,21,24)(18,22,19,23);(1,4,5,7)(2,8,6,3)(9,12)(10,11)(13,15)(14,16)Polycyclic presentation:
2^8|a^2=fg,c^2=f,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#40401, #7637716, #7638138, … |
| maximal normal subgroup of | C2×#40401, #7637716, #7638138, … |
| maximal quotient of | #522009, #522533, #523111, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33389);