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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(10,11)(14,15)(17,18)(19,21)(20,23)(22,24);(1,2,5,3)(4,7,8,6)(9,10,13,11)(12,15,16,14)(17,19,20,24)(18,21,23,22);(9,10,13,11)(12,14,16,15)(17,19)(18,22)(20,24)(21,23);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14);(1,4)(2,6)(3,7)(5,8)(9,12,13,16)(10,14,11,15)(17,18)(19,22)(20,23)(21,24)Polycyclic presentation:
2^8|b^2=fh,c^2=gh,d^2=f,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 | #6535282, #6535286, #6535288, … |
| maximal normal subgroup of | #6535282, #6535286, #6535288, … |
| maximal quotient of | #521481, #522185, #522729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32678);