Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,12,13)(11,15,16,14)(18,20)(22,23);(1,2,5,8)(3,7,4,6)(10,13)(14,15)(17,18,21,20)(19,23,24,22);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,20,23);(9,11,12,16)(10,14,13,15)(17,20,21,18)(19,23,24,22);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=gh,d^2=f,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 | #98403, #260485, #260489, … |
| maximal normal subgroup of | #98403, #260485, #260489, … |
| maximal quotient of | #521501, #522196, #522746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32629);