Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
746 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)(3,8)(4,7)(5,6)(9,10,13,12)(11,15,16,14)(18,22)(23,26)(24,29)(30,32);(1,3,6,4)(2,7,5,8)(18,22)(19,25)(20,28)(30,32);(3,4)(7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,4,6,3)(2,7,5,8)(9,12,13,10)(11,15,16,14)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32);(1,5,6,2)(3,7,4,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=f,c^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#50310, #7706211, #7706217, … |
| maximal normal subgroup of | C2×#50310, #7706211, #7706217, … |
| maximal quotient of | #561203, #561309, #562276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43298);