Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
975 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)(5,6,9,10)(7,11,15,13)(8,16,17,12)(14,19,20,18)(21,22,26,29)(23,24,27,31)(25,28,33,35)(30,32,34,36);(6,10)(8,17)(11,13)(14,20)(23,27)(24,31)(25,33)(28,35);(1,2)(3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23)(22,24)(25,30)(26,27)(28,32)(29,31)(33,34)(35,36);(5,7,9,15)(6,11,10,13)(8,14,17,20)(12,18,16,19)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,8,9,17)(6,12,10,16)(7,14,15,20)(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=h,d^2=f,e^2=f,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 | C2×#48788, #7704772, #7704953, … |
| maximal normal subgroup of | C2×#48788, #7704772, #7704953, … |
| maximal quotient of | #560332, #560351, #561436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41776);