Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 431 conjugacy classes of subgroups — too many to draw. Only its 119 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)(7,16)(8,13)(9,25)(10,14)(11,15)(12,18)(17,28)(19,33)(20,27)(21,36)(22,31)(23,29)(24,30)(26,34)(32,35);(5,7,9,18,11,19,10,20)(6,12,14,16,15,27,25,33)(8,21,22,17,24,34,23,35)(13,28,29,36,30,32,31,26);(5,8,11,24)(6,13,15,30)(7,17,19,35)(9,23,10,22)(12,26,27,36)(14,31,25,29)(16,32,33,28)(18,21,20,34)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=eh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11130, #357818, C4×#2009, … |
| maximal normal subgroup of | C2×#11130, #357818, C4×#2009, … |
| maximal quotient of | #116988, #117016, #117020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4118);