Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| 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 | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
492 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(1,2,3,4)(9,10)(11,19)(12,22)(13,24)(14,16)(15,21)(17,23)(18,20);(5,6)(7,8)(9,11,13,21,14,12,20,23)(10,15,18,19,16,17,24,22);(2,4)(6,8)(10,16)(11,12)(18,24)(21,23);(9,12,13,23,14,11,20,21)(10,17,18,22,16,15,24,19)Polycyclic presentation:
2^8|a^2=fh,b^2=e,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #501258, #502651, #503868, … |
| maximal normal subgroup of | #501258, #502651, #503868, … |
| maximal quotient of | #277356, #277362, #277368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17771);