Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
497 conjugacy classes of subgroups; only the 307 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 307 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,12,10,13,8,18)(7,14,17,19,15,20,16,11);(5,7,8,16,10,15,9,17)(6,11,12,14,13,19,18,20);(1,2,3,4);(7,15)(11,19)(14,20)(16,17)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #109823, #109892, #116573, … |
| maximal normal subgroup of | #109823, #109892, #116573, … |
| maximal quotient of | #191373, #191374, #191381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13794);