Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(12,16)(14,15)(17,18)(19,24)(20,22)(21,23);(2,3)(6,7)(10,11)(14,15)(18,20)(23,24);(9,10,13,11)(12,14,16,15)(17,19,22,21)(18,23,20,24);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,24,21,23);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,21,22,19)(18,24,20,23)Polycyclic presentation:
2^8|c^2=gh,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97280, #129617, #131128, … |
| maximal normal subgroup of | #97280, #129617, #131128, … |
| maximal quotient of | #553941, #554141, #554610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30237);