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) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(6,8)(9,10,13,14)(11,15,17,18)(12,20,21,16)(19,24,23,22);(2,4)(10,14)(11,17)(12,21)(22,24);(5,6)(7,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23);(1,2)(3,4)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23);(9,12,13,21)(10,16,14,20)(11,19,17,23)(15,22,18,24)Polycyclic presentation:
2^8|a^2=h,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 | #6993947, #6993956, #6993965, … |
| maximal normal subgroup of | #6993947, #6993956, #6993965, … |
| maximal quotient of | #535286, #535287, #535920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38079);