Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 8192 |
| 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 | |
301 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 301 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)(7,15,18,28)(8,12)(9,13)(10,14)(11,17,24,21)(16,31,33,30)(19,26)(20,27)(22,29)(23,34,25,32)(35,36);(2,4)(5,7,9,21,10,18,22,17)(6,11,13,28,14,24,29,15)(8,16,19,34,20,33,35,32)(12,25,26,30,27,23,36,31);(2,4)(5,8,9,19,10,20,22,35)(6,12,13,26,14,27,29,36)(7,16,21,34,18,33,17,32)(11,23,28,31,24,25,15,30)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,d^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dh,e^b=eh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11369, #358616, #358691, … |
| maximal normal subgroup of | C2×#11369, #358616, #358691, … |
| maximal quotient of | #121726, #121727, #121728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4357);