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 |
| prime radical | #2456 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,20)(15,21,22,16)(17,19,25,26)(23,27,28,24);(5,6,8,10)(7,9,12,11)(15,22)(16,21)(17,25)(19,26);(1,2)(3,4)(13,15,18,22)(14,16,20,21)(17,23,25,28)(19,24,26,27);(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,27)(26,28);(5,7,8,12)(6,9,10,11)(13,17)(14,19)(15,23)(16,24)(18,25)(20,26)(21,27)(22,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,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×#50176, #7706012, #7706049, … |
| maximal normal subgroup of | C2×#50176, #7706012, #7706049, … |
| maximal quotient of | #560330, #560371, #561426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43164);