Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 135 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)(9,10)(11,14)(12,15)(13,17)(16,20)(18,22)(19,21)(23,24);(1,2,4,3)(5,6,7,8)(9,11,13,18)(10,14,17,22)(12,19,20,24)(15,21,16,23);(6,8)(10,15)(12,20)(14,21)(16,17)(19,24)(22,23);(1,3,4,2)(11,18)(14,22)(19,24)(21,23)Polycyclic presentation:
2^8|b^2=egh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123171, #123181, #332583, … |
| maximal normal subgroup of | #123171, #123181, #332583, … |
| maximal quotient of | #317922, #317923, #317924, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21247);