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 | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
812 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)(11,15)(14,16)(17,18)(19,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32);(1,2,5,8)(3,7,4,6)(10,13)(11,15)(18,23)(19,26)(21,30)(28,31);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32);(1,4,5,3)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)Polycyclic presentation:
2^8|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 | #98471, #260732, #260734, … |
| maximal normal subgroup of | #98471, #260732, #260734, … |
| maximal quotient of | #557076, #557115, #557359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39006);