Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(11,15)(14,16)(17,18)(19,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32);(2,5)(3,7)(9,10,13,12)(11,14,15,16)(18,23)(19,26)(21,30)(28,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32);(1,3,4,7)(2,6,5,8)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,16,15,14)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)Polycyclic presentation:
2^8|b^2=fg,c^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46101, #7702606, #7702714, … |
| maximal normal subgroup of | C2×#46101, #7702606, #7702714, … |
| maximal quotient of | #557210, #557223, #557542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39089);