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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31);(2,4)(6,7)(9,10,13,12)(11,14,16,15)(18,20)(23,25)(24,28)(30,31);(1,2,5,4)(3,7,8,6)(9,11,13,16)(10,14,12,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32);(1,3,5,8)(2,6,4,7)(9,12,13,10)(11,15,16,14)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(1,4,5,2)(3,7,8,6)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=fgh,d^2=g,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×#50336, #7706049, #7706221, … |
| maximal normal subgroup of | C2×#50336, #7706049, #7706221, … |
| maximal quotient of | #561187, #561312, #562276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43324);