Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| 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 | |
1173 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)(13,14,19,21)(15,22,16,18)(17,25,27,20)(23,26,24,28);(5,6)(7,11)(8,9)(10,12)(13,15,19,16)(14,18,21,22)(17,23,27,24)(20,26,25,28);(1,2,3,4)(6,9)(10,11)(13,16,19,15)(14,18,21,22)(17,23,27,24)(20,28,25,26);(5,6)(7,10)(8,9)(11,12)(13,17,19,27)(14,20,21,25)(15,23,16,24)(18,26,22,28);(5,7)(6,10)(8,12)(9,11)(13,18)(14,15)(16,21)(17,26)(19,22)(20,23)(24,25)(27,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fg,d^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 | #482886, #6353122, #6353132, … |
| maximal normal subgroup of | #482886, #6353122, #6353132, … |
| maximal quotient of | #544525, #545085, #545500, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33583);