Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 131072 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
953 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 953 conjugacy classes of subgroups — too many to draw. Only its 247 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,16,14,17,12,18)(11,19,22,15,20,23,21,24);(1,2,3,4)(9,11)(10,15)(12,22)(13,21)(14,20)(16,19)(17,24)(18,23);(6,8)(10,16)(12,13)(15,19)(17,18)(21,22)(23,24);(10,17)(11,20)(16,18)(21,22)Polycyclic presentation:
2^8|a^2=g,b^2=eg,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=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 | #137690, #137774, #359611, … |
| maximal normal subgroup of | #137690, #137774, #359611, … |
| maximal quotient of | #193545, #193563, #193577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13921);