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 | |
1475 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)(9,10,14,17)(11,12,15,20)(13,21,23,16)(18,24,22,19);(2,4)(9,11,14,15)(10,12,17,20)(13,22,23,18)(16,24,21,19);(5,6,7,8)(9,11)(10,12)(13,18)(14,15)(16,19)(17,20)(21,24)(22,23);(1,2)(3,4)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,21)(23,24);(9,13,14,23)(10,16,17,21)(11,18,15,22)(12,19,20,24)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,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 | #6996482, #6996486, #6996489, … |
| maximal normal subgroup of | #6996482, #6996486, #6996489, … |
| maximal quotient of | #535336, #535380, #535969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38320);