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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
933 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)(21,22,26,24)(23,28,31,27)(25,32,34,29)(30,35,33,36);(6,10)(7,13)(8,16)(18,20)(21,23,26,31)(22,27,24,28)(25,33,34,30)(29,36,32,35);(1,2,3,4)(5,6)(7,14)(8,12)(9,10)(11,13)(15,20)(16,17)(18,19)(21,23)(22,28)(24,27)(25,30)(26,31)(29,36)(32,35)(33,34);(5,7)(6,11)(8,15)(9,13)(10,14)(12,18)(16,19)(17,20)(21,24,26,22)(23,28,31,27)(25,32,34,29)(30,36,33,35);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,25,26,34)(22,29,24,32)(23,30,31,33)(27,35,28,36)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=f,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 | C2×#51588, #7707635, #7707638, … |
| maximal normal subgroup of | C2×#51588, #7707635, #7707638, … |
| maximal quotient of | #560443, #560444, #561537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44576);