Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×#120 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
909 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 909 conjugacy classes of subgroups — too many to draw. Only its 239 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,10,14,11,13,9,12)(7,15,16,8,18,19,17,20);(1,2,3,4)(5,7)(6,8)(9,16)(10,17)(11,18)(12,19)(13,20)(14,15);(5,8)(6,7)(9,19)(10,15)(11,20)(12,16)(13,18)(14,17);(5,9,11,10)(6,12,13,14)(7,16,18,17)(8,19,20,15)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #146624, #146633, #146634, … |
| maximal normal subgroup of | #146624, #146633, #146634, … |
| maximal quotient of | #193788, #193797, #193810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13940);