Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(13,14,17,20)(15,21,22,28)(16,24,26,19)(18,25,27,23);(2,4)(13,15,17,22)(14,18,20,27)(16,25,26,23)(19,28,24,21);(5,7,8,11)(6,10,9,12)(15,22)(18,27)(21,28)(23,25)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=dh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101336, #101338, #165906, … |
| maximal normal subgroup of | #101336, #101338, #165906, … |
| maximal quotient of | #243332, #243333, #243334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16673);