Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,4)(6,7)(9,10,13,11)(12,15,16,14)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31);(1,2,5,8)(3,7,4,6)(9,11,13,10)(12,15,16,14)(18,20)(23,25)(24,28)(30,31);(1,3,5,4)(2,6,8,7)(9,12,13,16)(10,14,11,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(1,4,5,3)(2,7,8,6)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(1,2)(3,6)(4,7)(5,8)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=fg,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 | #98475, #260773, #260777, … |
| maximal normal subgroup of | #98475, #260773, #260777, … |
| maximal quotient of | #557083, #557113, #557360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39151);