Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
536 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,12)(7,11,8,10)(13,14,16,19)(15,23,25,28)(17,22,18,21)(20,24,27,26);(5,7,9,8)(6,10,12,11)(14,19)(15,24)(17,18)(20,23)(25,26)(27,28);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,15,18,24,16,25,17,26)(14,20,22,28,19,27,21,23);(5,8,9,7)(6,11,12,10)(14,19)(15,25)(21,22)(24,26)Polycyclic presentation:
2^8|a^2=e,b^2=eh,c^2=ef,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139232, #139292, #268058, … |
| maximal normal subgroup of | #139232, #139292, #268058, … |
| maximal quotient of | #304371, #304380, #304385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18933);