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 |
| prime radical | #2733 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,13)(12,15)(14,16)(18,22)(19,25)(24,29)(27,31);(2,3)(6,7)(10,11)(14,15)(20,28)(24,29)(27,31)(30,32);(9,10,13,11)(12,14,16,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,16)(10,14,11,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=fh,c^2=h,d^2=f,e^2=g,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×#56844, #8099980, #8100310, … |
| maximal normal subgroup of | C2×#56844, #8099980, #8100310, … |
| maximal quotient of | #586001, #586504, #586995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49832);