Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
737 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 737 conjugacy classes of subgroups — too many to draw. Only its 239 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:
(2,4)(5,6,10,14,11,15,9,25)(7,16,20,27,18,30,19,12)(8,21,24,28,22,35,23,13)(17,31,33,36,34,26,32,29);(1,2,3,4)(5,7)(6,12,15,27)(8,17)(9,19)(10,20)(11,18)(13,26,28,31)(14,16,25,30)(21,29,35,36)(22,34)(23,32)(24,33);(5,8,9,23,11,22,10,24)(6,13,14,21,15,28,25,35)(7,17,19,32,18,34,20,33)(12,26,16,29,27,31,30,36);(7,18)(8,22)(12,27)(13,28)(16,30)(19,20)(21,35)(23,24)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #124838, #140678, #357375, … |
| maximal normal subgroup of | #124838, #140678, #357375, … |
| maximal quotient of | #204269, #204270, #204286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14230);