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 | #2733 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(10,13)(14,16)(19,25)(22,26)(27,31)(30,32);(2,4)(6,7)(17,18,21,24)(19,26,25,22)(20,28,29,23)(27,30,31,32);(1,3)(2,6)(4,7)(5,8)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32);(1,3)(2,6)(4,7)(5,8)(17,20,21,29)(18,23,24,28)(19,27,25,31)(22,30,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=fg,d^2=gh,e^2=h,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 | #133706, #500515, #502806, … |
| maximal normal subgroup of | #133706, #500515, #502806, … |
| maximal quotient of | #557299, #557526, #557766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39311);