Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 524288 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1109 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)(5,6,9,8)(7,11,12,10)(16,23)(20,25)(22,27)(26,28);(5,7,9,12)(6,10,8,11)(14,18)(15,21)(20,25)(22,27);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #96979, #97162, #129824, … |
| maximal normal subgroup of | #96979, #97162, #129824, … |
| maximal quotient of | #516403, #516590, #516636, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28142);