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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,3)(4,7,5,8)(10,13)(14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32);(1,3,6,2)(4,7,5,8)(18,22)(23,25)(24,28)(30,31);(1,4,6,5)(2,7,3,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31);(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,5,6,4)(2,8,3,7)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=fh,d^2=fg,e^2=h,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×#39778, #7637338, #7637383, … |
| maximal normal subgroup of | C2×#39778, #7637338, #7637383, … |
| maximal quotient of | #521514, #522164, #522673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32766);