Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
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 | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
695 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 695 conjugacy classes of subgroups — too many to draw. Only its 263 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,19,17,15,16,20);(1,2,3,4)(6,9)(10,11)(13,15,17,14)(16,20,18,19);(5,7,8,12)(6,10,9,11)(13,16,17,18)(14,19,15,20);(13,16,17,18)(14,19,15,20)Polycyclic presentation:
2^8|a^2=g,b^2=efg,c^2=f,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^b=eh,g^b=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #107496, #107497, #110518, … |
| maximal normal subgroup of | #107496, #107497, #110518, … |
| maximal quotient of | #206709, #206718, #206721, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14301);