Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| 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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1125 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)(8,12)(10,11)(14,18)(15,21)(16,24)(26,28);(6,7)(10,11)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26);(1,2)(3,4)(5,6,9,7)(8,10,12,11);(5,7,9,6)(8,11,12,10)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28);(5,8)(6,10)(7,11)(9,12)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)Polycyclic presentation:
2^8|b^2=f,c^2=h,d^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132058, #133322, #213990, … |
| maximal normal subgroup of | #132058, #133322, #213990, … |
| maximal quotient of | #553720, #553978, #554196, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30221);