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 | #2661 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(7,8)(9,10,14,11)(12,15,13,16)(17,18,20,21)(19,23,24,22);(9,11,14,10)(12,15,13,16)(18,21)(22,23);(1,2)(3,7)(4,8)(5,6)(9,12,14,13)(10,15,11,16)(17,18)(19,22)(20,21)(23,24);(1,3,5,4)(2,7,6,8)(17,19)(18,22)(20,24)(21,23);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,16,11,15)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=fh,d^2=fg,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×#39833, #7637383, #7637417, … |
| maximal normal subgroup of | C2×#39833, #7637383, #7637417, … |
| maximal quotient of | #521517, #522167, #522660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32821);