Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×2−1+4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
493 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 493 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(1,2,4,3)(5,6,12,18)(7,19,8,22)(9,15,27,33)(10,17,11,16)(13,24,14,21)(20,35,23,31)(25,30,36,29)(26,34,28,32);(1,3,4,2)(5,7,12,8)(6,13,18,14)(9,25,27,36)(10,24,11,21)(15,31,33,35)(16,22,17,19)(20,28,23,26)(29,34,30,32);(5,8,12,7)(6,14,18,13)(9,23,27,20)(10,24,11,21)(15,30,33,29)(16,22,17,19)(25,26,36,28)(31,32,35,34);(5,9)(6,15)(7,20)(8,23)(10,28)(11,26)(12,27)(13,29)(14,30)(16,34)(17,32)(18,33)(19,35)(21,36)(22,31)(24,25)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,f^c=fh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152565, C2×#22892, #6381466, … |
| maximal normal subgroup of | #152565, C2×#22892, #6381466, … |
| maximal quotient of | #226936, #226944, #226952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15880);