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 | #2682 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,7,8,4)(10,11)(14,15);(2,6)(4,7)(9,10,13,11)(12,15,16,14)(18,22)(20,23);(9,10,13,11)(12,14,16,15)(17,18)(19,23)(20,24)(21,22);(1,3)(2,4)(5,8)(6,7)(9,11,13,10)(12,15,16,14)(17,19,21,24)(18,23,22,20);(1,4)(2,3)(5,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^8|a^2=fg,b^2=g,c^2=g,d^2=gh,e^2=h,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×#40205, #7637518, #7638132, … |
| maximal normal subgroup of | C2×#40205, #7637518, #7638132, … |
| maximal quotient of | #522001, #522525, #523103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33193);