Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| 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 | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,3)(4,7,5,8)(9,10,13,11)(12,15,16,14)(17,18,20,22)(19,23,24,21);(1,3,6,2)(4,7,5,8)(10,11)(14,15)(17,19,20,24)(18,21,22,23);(1,4,6,5)(2,7,3,8)(9,11,13,10)(12,15,16,14)(17,19)(18,21)(20,24)(22,23);(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,22)(23,24);(1,5,6,4)(2,8,3,7)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=gh,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 | #97849, #97956, #166424, … |
| maximal normal subgroup of | #97849, #97956, #166424, … |
| maximal quotient of | #521497, #522150, #522654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32748);