Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×2−1+4 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1797 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 1797 conjugacy classes of subgroups — too many to draw. Only its 303 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 — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,13,22,27)(7,15,17,28,18,11,21,24)(8,12,20,26,19,25,32,35)(16,29,31,36,30,23,33,34);(5,7,10,18)(6,11,13,15)(8,16,19,30)(9,21,22,17)(12,23,25,29)(14,28,27,24)(20,33,32,31)(26,36,35,34);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,25)(14,26)(15,29)(17,31)(18,30)(21,33)(22,32)(24,34)(27,35)(28,36);(8,19)(12,25)(16,30)(20,32)(23,29)(26,35)(31,33)(34,36)Polycyclic presentation:
2^8|a^2=g,b^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #128066, #128068, #128072, … |
| maximal normal subgroup of | #128066, #128068, #128072, … |
| maximal quotient of | #184749, #184751, #184752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13581);