Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2431 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,24,23,20);(3,8)(6,7)(10,14)(11,16)(18,22)(19,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,23)(18,20,22,24);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,23)(19,22)(21,24);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,15)(14,16)(17,20)(18,23)(19,22)(21,24)Polycyclic presentation:
2^8|a^2=fh,c^2=gh,e^2=fg,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 | #7007266, #7007286, #7007319, … |
| maximal normal subgroup of | #7007266, #7007286, #7007319, … |
| maximal quotient of | #531791, #531941, #532148, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52742);