Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61109 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1300 conjugacy classes of subgroups; only the 485 normal subgroups are shown.
This group has 1300 conjugacy classes of subgroups — too many to draw. Only its 485 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,16)(5,9)(6,23)(7,10)(8,15)(11,24)(13,30)(14,18)(17,19)(20,29)(21,28)(22,27)(25,32)(26,31);(1,3)(2,8)(4,10)(5,19)(6,16)(7,14)(9,26)(11,18)(12,23)(13,20)(15,24)(17,29)(21,31)(22,32)(25,28)(27,30);(1,4,7,8)(2,3,10,14)(5,17,22,25)(6,18,24,12)(9,13,27,31)(11,15,23,16)(19,29,32,28)(20,30,21,26);(1,5)(2,9)(3,13)(4,17)(6,21)(7,22)(8,25)(10,27)(11,29)(12,30)(14,31)(15,32)(16,19)(18,26)(20,24)(23,28);(2,10)(3,14)(9,27)(11,23)(13,31)(15,16)(19,32)(28,29)Polycyclic presentation:
2^8|c^2=h,f^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gh,d^b=df,e^b=eh,f^b=fh,g^c=gh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152181, #152193, #152195, … |
| maximal normal subgroup of | #152181, #152193, #152195, … |
| maximal quotient of | #6358844, #6358847, #6358853, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54099);