Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
845 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 845 conjugacy classes of subgroups — too many to draw. Only its 247 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,4)(5,6)(7,8)(9,10,14,20,15,18,13,19)(11,21,25,16,23,31,24,32)(12,26,28,17,29,34,27,33)(22,30,38,35,36,40,37,39);(1,2,3,4)(9,11)(10,16)(12,22)(13,25)(14,24)(15,23)(17,30)(18,32)(19,31)(20,21)(26,35)(27,38)(28,37)(29,36)(33,40)(34,39);(6,8)(9,12,15,29)(10,17,18,33)(11,22,23,36)(13,28,14,27)(16,30,32,40)(19,34,20,26)(21,35,31,39)(24,38,25,37);(10,18)(11,23)(17,33)(19,20)(22,36)(24,25)(26,34)(37,38)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #137691, #137775, #359612, … |
| maximal normal subgroup of | #137691, #137775, #359612, … |
| maximal quotient of | #193547, #193565, #193579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13923);