Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
563 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 563 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10)(11,19)(12,18)(13,15)(14,17)(16,20)(21,24)(22,23);(1,2,4,3)(5,7,8,6)(10,15)(16,23)(17,18)(19,24);(1,3,4,2)(9,11,13,21)(10,16,15,23)(12,20,14,22)(17,19,18,24);(9,12,13,14)(10,17,15,18)(11,20,21,22)(16,19,23,24)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #100012, #100013, #100028, … |
| maximal normal subgroup of | #100012, #100013, #100028, … |
| maximal quotient of | #202323, #202338, #202353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14049);