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 | #2670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(18,22)(23,24);(1,2,5,3)(4,7,8,6)(10,13)(11,15);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22);(1,3,5,2)(4,7,8,6)(17,19,21,20)(18,23,22,24);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=g,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×#39090, #7636408, #7636571, … |
| maximal normal subgroup of | C2×#39090, #7636408, #7636571, … |
| maximal quotient of | #521588, #522287, #522844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32078);