Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,4,5,8)(11,15)(13,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30);(2,7)(3,5)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31);(1,3,6,5)(2,8,7,4)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32);(1,4)(2,3)(5,7)(6,8)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,5,6,3)(2,4,7,8)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=f,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97353, #130141, #130152, … |
| maximal normal subgroup of | #97353, #130141, #130152, … |
| maximal quotient of | #557046, #557388, #557407, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39133);