Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | |
961 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)(5,6,10,8)(7,11,13,15)(9,12,17,16)(14,18,19,20)(21,22)(23,25)(24,28)(26,34)(27,30)(29,35)(31,36)(32,33);(7,13)(9,17)(11,15)(12,16)(21,23,27,24)(22,28,30,25)(26,31,35,32)(29,36,34,33);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,20)(17,19)(21,24,27,23)(22,28,30,25)(26,32,35,31)(29,36,34,33);(5,8,10,6)(7,11,13,15)(9,16,17,12)(14,18,19,20)(21,25,27,28)(22,23,30,24)(26,33,35,36)(29,31,34,32);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)(21,26,27,35)(22,29,30,34)(23,31,24,32)(25,33,28,36)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,d^2=fh,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47771, #7701775, #7701779, … |
| maximal normal subgroup of | C2×#47771, #7701775, #7701779, … |
| maximal quotient of | #576641, #576698, #576731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40759);