Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(4,8)(6,7)(10,14)(12,15)(17,18,21,24)(19,25,27,22)(20,23,28,30)(26,31,32,29);(2,3)(6,7)(10,14)(11,16)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,15,14,12)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,15)(10,16,14,11)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,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 | #97286, #168063, #168064, … |
| maximal normal subgroup of | #97286, #168063, #168064, … |
| maximal quotient of | #553941, #553942, #554136, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30240);