Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
858 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(9,11,12,16)(10,13,14,15)(18,22)(23,28)(24,29)(30,32);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,25)(23,28)(26,31)(30,32);(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)Polycyclic presentation:
2^8|b^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133174, #251818, #251819, … |
| maximal normal subgroup of | #133174, #251818, #251819, … |
| maximal quotient of | #544274, #544278, #544289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29680);