Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
327 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,13)(8,16)(9,12)(10,15)(11,17)(14,19)(18,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(5,7)(6,10,12,18)(8,15)(9,14)(11,13,16,19)(17,20)(21,23)(22,26,28,29)(24,32)(25,31)(27,35,36,34)(30,33);(7,14)(10,18)(13,19)(15,20)(21,24,25,33)(22,27,28,36)(23,30,31,32)(26,34,29,35)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=g,b^a=bd,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #106043, #107308, #107309, … |
| maximal normal subgroup of | #106043, #107308, #107309, … |
| maximal quotient of | #95207, #95211, #95214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1013);