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 | |
946 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)(5,6)(7,15)(8,17)(9,14)(10,11)(12,18)(13,19)(16,20)(21,22,26,29)(23,24,27,32)(25,28,34,36)(30,31,33,35);(6,11)(9,20)(12,15)(13,17)(21,23,26,27)(22,24,29,32)(25,33,34,30)(28,35,36,31);(1,2,3,4)(5,7,10,18)(6,12,11,15)(8,14,19,16)(9,17,20,13)(21,23)(22,24)(25,30)(26,27)(28,31)(29,32)(33,34)(35,36);(5,8,10,19)(6,13,11,17)(7,16,18,14)(9,12,20,15)(21,24)(22,27)(23,29)(25,31)(26,32)(28,33)(30,36)(34,35);(5,9,10,20)(6,14,11,16)(7,17,18,13)(8,12,19,15)(21,25,26,34)(22,28,29,36)(23,30,27,33)(24,31,32,35)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,d^2=f,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 | #132399, #248440, #248441, … |
| maximal normal subgroup of | #132399, #248440, #248441, … |
| maximal quotient of | #544520, #545080, #545508, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35872);