Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 599 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,20,18,22,16,21)(15,19,24,27,25,28,23,26);(5,7,8,12)(6,9,10,11)(13,15,17,24,18,25,16,23)(14,19,20,27,22,28,21,26);(2,4)(14,20)(15,23)(16,17)(21,22)(24,25)(26,27);(6,10)(9,11)Polycyclic presentation:
2^8|a^2=fh,b^2=eg,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #301885, #301890, #301946, … |
| maximal normal subgroup of | #301885, #301890, #301946, … |
| maximal quotient of | #319006, #319017, #319018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21318);