Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
780 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 163 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)(9,10,13,16)(11,17,18,24)(12,20,21,15)(14,22,23,19);(1,2,4,3)(5,6)(7,8);(6,8)(9,11,12,19,13,18,21,22)(10,14,15,17,16,23,20,24);(1,3,4,2)(11,18)(14,23)(17,24)(19,22)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #175374, #175380, #175398, … |
| maximal normal subgroup of | #175374, #175380, #175398, … |
| maximal quotient of | #297264, #297271, #297278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18714);