Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 262144 |
| 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 | |
1011 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,20,22)(19,23,24,21);(2,5)(6,7)(10,11)(14,15)(17,18)(19,21)(20,22)(23,24);(9,10,13,11)(12,14,16,15);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14);(1,3,4,8)(2,6,5,7)(9,12,13,16)(10,14,11,15)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=fh,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #98637, #130136, #251576, … |
| maximal normal subgroup of | #98637, #130136, #251576, … |
| maximal quotient of | #531695, #531872, #532402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52291);