Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(1,2,3,4)(5,6,9,12)(7,11,8,10)(13,14,17,20)(15,18,23,27)(16,19,24,28)(21,25,22,26);(5,7,9,8)(6,10,12,11)(13,15,17,23)(14,18,20,27)(16,21,24,22)(19,25,28,26);(2,4)(15,21)(16,24)(18,25)(19,28)(22,23)(26,27);(5,8,9,7)(6,11,12,10)Polycyclic presentation:
2^8|a^2=efh,b^2=eh,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #269729, #269746, #269759, … |
| maximal normal subgroup of | #269729, #269746, #269759, … |
| maximal quotient of | #279261, #279262, #279263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17802);