Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| 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 | #2414 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
867 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10,14,11)(12,15,13,16)(18,21)(22,24);(2,4)(6,7)(9,11,14,10)(12,15,13,16);(9,12,14,13)(10,15,11,16)(17,18)(19,22)(20,21)(23,24);(1,3)(2,6)(4,7)(5,8)(17,19,20,23)(18,22,21,24);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,16,11,15)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=h,d^2=g,e^2=fh,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 | #6697852, #6697857, #6697917, … |
| maximal normal subgroup of | #6697852, #6697857, #6697917, … |
| maximal quotient of | #521506, #522158, #522665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32769);