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) = C28 |
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 | |
1155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,19)(15,16,20,24)(17,25,26,21)(22,28,27,23);(6,9)(10,11)(14,19)(15,20)(17,26)(23,28);(1,2,3,4)(13,15)(14,16)(17,22)(18,20)(19,24)(21,23)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,19)(17,23)(18,24)(21,27)(22,25)(26,28);(5,7,8,12)(6,10,9,11)(13,17,18,26)(14,21,19,25)(15,22,20,27)(16,23,24,28)Polycyclic presentation:
2^8|a^2=h,c^2=g,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×#48821, #7704557, #7704955, … |
| maximal normal subgroup of | C2×#48821, #7704557, #7704955, … |
| maximal quotient of | #560418, #560449, #561521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41809);