Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 |
| prime radical | #2541 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1008 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)(3,5)(4,7)(6,8)(9,10,13,14)(11,16,12,15)(17,18,20,22)(19,23,24,21);(2,3)(6,7)(10,14)(15,16)(17,18)(19,21)(20,22)(23,24);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14);(1,3,5,2)(4,7,8,6)(9,11,13,12)(10,15,14,16);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,16,14,15)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=fh,c^2=h,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 | #98684, #258887, #258889, … |
| maximal normal subgroup of | #98684, #258887, #258889, … |
| maximal quotient of | #531817, #531820, #531890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52628);