Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1071 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 1071 conjugacy classes of subgroups — too many to draw. Only its 191 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,17)(11,15,18,22)(12,14,20,16)(19,21,24,23);(5,6)(7,8)(10,14)(12,20)(15,21)(16,17)(19,24)(22,23);(2,4)(6,8)(9,11)(10,15)(12,19)(13,18)(14,21)(16,23)(17,22)(20,24);(11,18)(15,22)(19,24)(21,23)Polycyclic presentation:
2^8|a^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,c^b=cg,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #394057, #394083, #394084, … |
| maximal normal subgroup of | #394057, #394083, #394084, … |
| maximal quotient of | #232954, #232955, #232956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16376);