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 | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(3,7)(10,15)(11,13)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(1,2,5,4)(3,8,7,6)(9,10,14,15)(11,12,13,16)(18,22)(19,26)(24,29)(27,32);(3,7)(6,8)(9,11,14,13)(10,16,15,12)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,3,5,7)(2,6,4,8)(9,12,14,16)(10,11,15,13)(17,19,21,26)(18,23,22,25)(20,27,28,32)(24,30,29,31);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,16,15,12)(17,20,21,28)(18,24,22,29)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|b^2=fgh,c^2=g,d^2=f,e^2=f,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 | #132559, #487320, #487894, … |
| maximal normal subgroup of | #132559, #487320, #487894, … |
| maximal quotient of | #582254, #582488, #582645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45736);