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 | |
761 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(18,22)(23,26)(24,29)(30,32);(1,3,5,8)(2,4,6,7)(18,22)(19,25)(20,28)(30,32);(1,2,5,6)(3,7,8,4)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,2)(3,4)(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,4)(2,3)(5,7)(6,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=gh,b^2=f,c^2=f,d^2=fg,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×#48953, #7704502, #7704961, … |
| maximal normal subgroup of | C2×#48953, #7704502, #7704961, … |
| maximal quotient of | #561178, #561210, #562381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41941);