Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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×#535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1251 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)(5,6)(7,8)(9,10)(11,12)(13,14,17,18)(15,19,23,21)(16,24,25,20)(22,27,28,26);(6,10)(7,11)(14,18)(16,25)(19,21)(22,28);(1,2)(3,4)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(5,7,9,11)(6,8,10,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(5,8)(6,11)(7,10)(9,12)(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=h,d^2=fh,e^2=h,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 | #131980, #133351, #133357, … |
| maximal normal subgroup of | #131980, #133351, #133357, … |
| maximal quotient of | #560365, #561461, #562643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45042);