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 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
796 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)(3,5)(4,7)(6,8)(11,15)(14,16)(17,18,22,24)(19,25,27,20)(21,29,30,23)(26,28,32,31);(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,20,24,25)(21,26,30,32)(23,28,29,31);(2,3)(6,7)(10,13)(14,16)(17,20,22,25)(18,19,24,27)(21,28,30,31)(23,26,29,32);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=g,e^2=fgh,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 | #7221588, #7221592, #7221652, … |
| maximal normal subgroup of | #7221588, #7221592, #7221652, … |
| maximal quotient of | #582292, #582494, #582533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45989);