Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 131072 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(21,22,27,26)(23,30,24,28)(25,33,34,29)(31,35,32,36);(6,10)(8,16)(11,13)(14,20)(21,23,27,24)(22,28,26,30)(25,31,34,32)(29,35,33,36);(1,2)(3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,24,27,23)(22,28,26,30)(25,31,34,32)(29,36,33,35);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,25,27,34)(22,29,26,33)(23,31,24,32)(28,35,30,36);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,26,27,22)(23,28,24,30)(25,33,34,29)(31,35,32,36)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132348, #248816, #248817, … |
| maximal normal subgroup of | #132348, #248816, #248817, … |
| maximal quotient of | #544426, #545012, #545366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35903);