Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2011 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
404 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10,12,15,14,18,13,17)(11,19,20,24,22,23,21,16);(2,5)(6,7)(9,10)(11,16)(12,17)(13,15)(14,18)(19,21)(20,23)(22,24);(10,15)(12,13)(16,23)(17,18)(19,24)(20,21);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,21)(13,20)(14,22)(15,23)(17,19)(18,24)Polycyclic presentation:
2^8|a^2=e,d^2=ef,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh,e^d=eh,f^d=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #151228, #151230, #151412, … |
| maximal normal subgroup of | #151228, #151230, #151412, … |
| maximal quotient of | #312025, #312026, #312027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19908);