Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
286 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(2,4)(5,6,9,13,10,14,8,12)(7,15,16,11,18,19,17,20)(21,22)(23,26)(24,31,32,27)(25,28)(29,33,35,36)(30,34);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,19)(13,15)(14,20)(21,23,25,30)(22,26,28,34)(24,29,32,35)(27,33,31,36);(5,8,10,9)(6,12,14,13)(7,16,18,17)(11,19,20,15)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,35)(34,36)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,c^2=g,d^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #132950, #132951, #132953, … |
| maximal normal subgroup of | #132950, #132951, #132953, … |
| maximal quotient of | #99556, #99573, #99646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2010);