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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(10,13)(14,16)(17,18,22,25)(19,20,23,27)(21,24,29,31)(26,28,30,32);(1,3,6,4)(2,7,8,5)(19,23)(20,27)(21,29)(24,31);(2,8)(3,4)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,26)(22,23)(24,28)(25,27)(29,30)(31,32);(1,4,6,3)(2,7,8,5)(9,11,12,15)(10,14,13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(1,5,6,7)(2,3,8,4)(17,21,22,29)(18,24,25,31)(19,26,23,30)(20,28,27,32)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,d^2=g,e^2=h,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×#51849, #7707768, #7707849, … |
| maximal normal subgroup of | C2×#51849, #7707768, #7707849, … |
| maximal quotient of | #561299, #561316, #562327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44837);