Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(25,33)(28,32)(30,36)(34,35);(6,7)(11,13)(12,14)(18,19)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(1,2)(3,4)(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,24,26,22)(23,27,31,29)(25,32,33,28)(30,34,36,35);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^8|a^2=fg,b^2=fh,c^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #112002, #112038, #112064, … |
| maximal normal subgroup of | #112002, #112038, #112064, … |
| maximal quotient of | #556347, #556378, #556393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30623);