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:
(1,2,6,7)(3,5,4,8)(17,18)(19,24)(20,26)(21,23)(22,25)(27,31)(28,32)(29,30);(2,7)(3,4)(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,24,21,26)(22,28,30,27)(25,32,29,31);(1,3,6,4)(2,8,7,5)(9,10,12,14)(11,13,16,15)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32);(1,4,6,3)(2,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31);(1,5)(2,3)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,22,23,30)(18,25,21,29)(19,27,20,28)(24,31,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=fgh,d^2=g,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×#56604, #8099053, #8099293, … |
| maximal normal subgroup of | C2×#56604, #8099053, #8099293, … |
| maximal quotient of | #586343, #586407, #588495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49592);