Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| 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 | #117 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #382 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 295 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)(7,15,18,11)(8,19,21,34)(9,14)(10,13)(12,20,25,35)(16,29)(17,28,32,24)(22,26)(23,30)(27,36)(31,33);(1,2)(3,4)(5,7,9,17,10,18,22,32)(6,11,14,24,13,15,26,28)(8,16,20,30,21,31,35,36)(12,23,19,33,25,27,34,29);(2,4)(5,8)(6,12,13,25)(7,16)(9,20)(10,21)(11,23,15,27)(14,19,26,34)(17,30)(18,31)(22,35)(24,33,28,29)(32,36)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,e^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11259, #358268, #358418, … |
| maximal normal subgroup of | C2×#11259, #358268, #358418, … |
| maximal quotient of | #120677, #120678, #120702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4247);