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 | |
925 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 925 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)(9,10,14,18,15,11,13,17)(12,20,22,16,23,24,21,19);(1,2,3,4)(9,11,15,10)(12,19,23,16)(13,17,14,18)(20,22,24,21);(6,8)(9,12,15,23)(10,16,11,19)(13,22,14,21)(17,24,18,20);(10,11)(16,19)(17,18)(20,24)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=h,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 | #137689, #137773, #359610, … |
| maximal normal subgroup of | #137689, #137773, #359610, … |
| maximal quotient of | #193548, #193566, #193580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13924);