Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | C2×#540 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1028 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)(5,6,10,7)(8,11,16,13)(9,14,17,12)(15,19,20,18)(21,22,26,30)(23,24,28,27)(25,29,33,36)(31,32,35,34);(6,7)(9,17)(11,13)(15,20)(23,28)(24,27)(25,33)(29,36);(1,2)(3,4)(5,7,10,6)(8,11,16,13)(9,14,17,12)(15,18,20,19)(21,23)(22,27)(24,30)(25,31)(26,28)(29,34)(32,36)(33,35);(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,19)(17,20)(21,24,26,27)(22,28,30,23)(25,32,33,34)(29,35,36,31);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,25,26,33)(22,29,30,36)(23,31,28,35)(24,32,27,34)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=fh,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132094, #246934, #246961, … |
| maximal normal subgroup of | #132094, #246934, #246961, … |
| maximal quotient of | #560366, #561462, #562644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45045);