Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
858 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)(3,8)(4,6)(5,7)(9,10,13,15)(11,16,12,14);(1,3,5,4)(2,6,7,8)(17,18)(19,22)(20,21)(23,24);(9,11,13,12)(10,14,15,16)(18,21)(22,24);(1,4,5,3)(2,6,7,8)(18,21)(22,24);(9,12,13,11)(10,14,15,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=g,d^2=f,e^2=gh,b^a=bf,c^a=cg,c^b=ch,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #133246, #251812, #251813, … |
| maximal normal subgroup of | #133246, #251812, #251813, … |
| maximal quotient of | #514236, #514516, #514520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27961);