Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13334 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2011 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
284 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 3 generators on 32 points:
(1,2)(3,14)(4,8)(5,21)(6,24)(7,12)(9,15)(10,18)(11,22)(13,20)(16,29)(17,26)(19,28)(23,31)(25,30)(27,32);(2,8)(4,18)(5,22)(6,20)(9,14)(10,13)(11,24)(12,21)(15,23)(16,25)(19,32)(26,27)(28,29)(30,31);(1,3)(2,9)(4,19)(5,23)(6,25)(7,17)(8,26)(10,29)(11,30)(12,27)(13,31)(14,21)(15,22)(16,20)(18,32)(24,28)Polycyclic presentation:
2^8|d^2=gh,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eg,g^b=gh,d^c=dgh,e^c=eg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13334, #498872, #499269, … |
| maximal normal subgroup of | C2×#13334, #498872, #499269, … |
| maximal quotient of | #142624, #142625, #142628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6322);