Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
884 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)(3,7)(4,5)(6,8)(10,11)(14,15)(18,22)(19,25)(20,28)(30,32);(2,5)(3,8)(9,10,13,11)(12,15,16,14);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30);(1,3)(2,6)(4,8)(5,7)(9,12,13,16)(10,14,11,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|b^2=fh,c^2=g,d^2=g,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #133018, #487140, #487268, … |
| maximal normal subgroup of | #133018, #487140, #487268, … |
| maximal quotient of | #582154, #582366, #582592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45787);