Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,16,14,18,12,17)(11,19,20,15,22,23,21,24);(1,2,3,4)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,23)(18,24);(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,23)(18,24);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,24,23)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100011, #100014, #100027, … |
| maximal normal subgroup of | #100011, #100014, #100027, … |
| maximal quotient of | #193539, #193557, #193570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13915);