Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
267 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 267 conjugacy classes of subgroups — too many to draw. Only its 57 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:
(2,8)(3,13)(4,17)(5,6)(9,26)(10,24)(11,18)(12,25)(14,27)(15,19)(16,22)(20,21)(23,28)(29,32);(1,2)(3,14)(4,18)(5,10)(6,11)(7,12)(8,17)(9,13)(15,23)(16,28)(19,27)(20,25)(21,24)(22,29)(26,30)(31,32);(1,3)(2,9)(4,19)(5,22)(6,13)(7,16)(8,23)(10,14)(11,28)(12,29)(15,21)(17,31)(18,32)(20,30)(24,26)(25,27)Polycyclic presentation:
2^8|d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,f^b=fh,g^b=gh,d^c=dh,e^c=eh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13323, #498837, #499514, … |
| maximal normal subgroup of | C2×#13323, #498837, #499514, … |
| maximal quotient of | #142246, #142252, #142254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6311);