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 | #2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(9,10)(11,16)(12,15)(13,14)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31);(1,2,5,4)(3,7,8,6)(10,14)(15,16)(18,23)(19,26)(21,30)(28,31);(9,11,13,12)(10,15,14,16)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=g,d^2=fg,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×#49097, #7704878, #7704966, … |
| maximal normal subgroup of | C2×#49097, #7704878, #7704966, … |
| maximal quotient of | #561290, #561379, #562329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42085);