Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #399 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 111 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)(7,16,20,32)(8,21)(9,14)(10,15)(11,25)(12,18,28,19)(13,24)(17,33,34,29)(22,36,35,27)(23,31)(26,30);(2,4)(5,7,11,18,10,20,9,19)(6,12,25,16,15,28,14,32)(8,22,26,34,24,35,23,17)(13,29,31,27,21,33,30,36);(2,4)(5,8,11,26,10,24,9,23)(6,13,25,31,15,21,14,30)(7,17,18,22,20,34,19,35)(12,27,16,33,28,36,32,29)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,d^2=h,f^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93985, #137451, #137452, … |
| maximal normal subgroup of | #93985, #137451, #137452, … |
| maximal quotient of | #121624, #121667, #121668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4362);