Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
714 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 714 conjugacy classes of subgroups — too many to draw. Only its 145 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:
(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,17,25)(16,23,24,27)(20,26,28,22);(1,2)(3,4)(6,9)(10,12)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,26)(18,24)(19,27)(21,28)(23,25);(6,9)(10,12)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101520, #101522, #291740, … |
| maximal normal subgroup of | #101520, #101522, #291740, … |
| maximal quotient of | #254571, #254580, #254589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16998);