Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #386 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
411 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 411 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:
(5,6)(7,17)(8,13,24,32)(9,16)(10,14)(11,15)(12,19)(18,29,35,33)(20,27)(21,28)(22,26,34,36)(23,31,25,30);(1,2)(3,4)(5,7)(6,12)(8,22)(9,21)(10,19)(11,20)(13,29)(14,27)(15,28)(16,17)(18,25)(23,35)(24,34)(26,31)(30,36)(32,33);(2,4)(5,8,9,23,11,24,10,25)(6,13,14,30,15,32,16,31)(7,18,19,34,20,35,21,22)(12,26,17,33,28,36,27,29)Polycyclic presentation:
2^8|a^2=g,c^2=d,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,e^c=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11194, #357969, #358190, … |
| maximal normal subgroup of | C2×#11194, #357969, #358190, … |
| maximal quotient of | #118693, #118817, #118821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4182);