Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,22)(16,19)(17,25)(18,21)(20,26)(23,27)(24,28);(5,6)(7,11)(8,9)(10,12)(13,15,18,16)(14,19,21,22)(17,23,26,24)(20,27,25,28);(1,2)(3,4)(6,9)(10,11)(13,16,18,15)(14,19,21,22)(17,23,26,24)(20,28,25,27);(5,6)(7,10)(8,9)(11,12)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,27)(25,26)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#40511, #7638843, #7639570, … |
| maximal normal subgroup of | C2×#40511, #7638843, #7639570, … |
| maximal quotient of | #544419, #545007, #545359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33499);