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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
848 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)(3,4)(5,6)(7,8)(10,11)(14,15);(2,6)(4,7)(9,10,13,11)(12,15,16,14)(18,21)(22,24);(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,21)(23,24);(1,3)(2,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(17,19,20,23)(18,22,21,24);(1,4,5,7)(2,3,6,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=gh,e^2=fg,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×#38884, #7636057, #7636326, … |
| maximal normal subgroup of | C2×#38884, #7636057, #7636326, … |
| maximal quotient of | #521854, #522422, #523101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31872);