Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
789 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)(11,15,16,13)(18,22)(19,25)(24,29)(27,31);(1,3,5,8)(2,4,6,7)(9,11,12,16)(10,13,14,15)(20,28)(24,29)(27,31)(30,32);(2,6)(4,7)(9,11)(10,13)(12,16)(14,15)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#54724, #8102312, #8102632, … |
| maximal normal subgroup of | C2×#54724, #8102312, #8102632, … |
| maximal quotient of | #586114, #586576, #586937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47712);