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 | |
796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,7,5,8)(10,13)(11,15)(17,18)(19,25)(20,22)(21,28)(23,26)(24,29)(27,30)(31,32);(2,4)(3,5)(10,13)(14,16)(18,20)(19,26)(24,28)(27,32);(1,3,6,5)(2,7,4,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,4,6,2)(3,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30);(1,5,6,3)(2,7,4,8)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=gh,c^2=fg,d^2=f,e^2=f,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×#54045, #8101675, #8101945, … |
| maximal normal subgroup of | C2×#54045, #8101675, #8101945, … |
| maximal quotient of | #585601, #585610, #586457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47033);