Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2410 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
661 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 661 conjugacy classes of subgroups — too many to draw. Only its 133 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 — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,16,17,8)(5,10)(6,9)(12,19,23,15)(13,22)(18,29,30,24)(20,21)(25,28)(26,31)(27,32);(1,3,6,14)(2,7,9,11)(4,13,17,28)(5,19,21,15)(8,22,16,25)(10,12,20,23)(18,31,30,27)(24,26,29,32);(1,2)(3,12)(4,8)(5,20)(6,9)(7,19)(10,21)(11,15)(13,26)(14,23)(16,17)(18,29)(22,31)(24,30)(25,27)(28,32);(1,4)(2,8)(3,13)(5,18)(6,17)(7,22)(9,16)(10,24)(11,25)(12,26)(14,28)(15,27)(19,31)(20,29)(21,30)(23,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gh,e^c=eh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501109, #501112, #501115, … |
| maximal normal subgroup of | #501109, #501112, #501115, … |
| maximal quotient of | #267373, #267378, #267379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17398);