Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10,14,11)(12,15,13,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32);(9,11,14,10)(12,15,13,16)(18,22)(23,29)(24,27)(30,31);(1,2)(3,7)(4,8)(5,6)(9,12,14,13)(10,15,11,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(1,3,5,4)(2,7,6,8)(17,20)(18,24)(19,25)(21,28)(22,27)(23,30)(26,32)(29,31);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,16,11,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=fh,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#39826, #7637338, #7637426, … |
| maximal normal subgroup of | C2×#39826, #7637338, #7637426, … |
| maximal quotient of | #521524, #522172, #522671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32814);