Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | C22×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
513 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 513 conjugacy classes of subgroups — too many to draw. Only its 175 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:
(2,4)(5,6,8,7)(9,10,14,15,13,16,12,18)(11,19,22,24,21,17,20,23);(5,7,8,6)(10,15)(11,20)(12,14)(16,18)(21,22)(23,24);(1,2,3,4)(10,16)(15,18)(17,19)(23,24);(9,11,12,20,13,21,14,22)(10,17,18,24,16,19,15,23)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=fh,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #498333, #498466, #498580, … |
| maximal normal subgroup of | #498333, #498466, #498580, … |
| maximal quotient of | #231413, #231426, #231427, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16177);