Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 | #2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1141 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)(5,6,9,11)(7,8,12,10)(13,14,17,18)(15,19,21,22)(16,24,25,20)(23,28,27,26);(5,7,9,12)(6,10,11,8)(14,18)(15,21)(16,25)(26,28);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27);(5,7)(6,8)(9,12)(10,11)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,28)(25,27);(13,16,17,25)(14,20,18,24)(15,23,21,27)(19,26,22,28)Polycyclic presentation:
2^8|a^2=f,b^2=fh,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6994467, #6994471, #6994531, … |
| maximal normal subgroup of | #6994467, #6994471, #6994531, … |
| maximal quotient of | #535286, #535293, #535920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38225);