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 | #2414 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1032 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)(5,6,10,11)(7,12,8,9)(13,14,18,15)(16,19,23,21)(17,22,25,20)(24,27,28,26);(5,7,10,8)(6,9,11,12)(14,15)(16,23)(17,25)(26,27);(1,2)(3,4)(5,8,10,7)(6,9,11,12)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27);(5,9)(6,7)(8,11)(10,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28);(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=f,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 | #6998999, #6999003, #6999063, … |
| maximal normal subgroup of | #6998999, #6999003, #6999063, … |
| maximal quotient of | #535309, #535582, #535934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38227);