Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | Aut(G) = #72338915 (order 4096) · Inn(G) = #2826 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2826 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
251 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,5,9,7,12,6,11)(3,13,16,8,17,23,14,24)(4,10,20,25,21,26,19,27)(15,28,30,22,18,31,29,32);(1,3,5,16,7,17,6,14)(2,8,11,13,12,24,9,23)(4,18,19,30,21,15,20,29)(10,22,25,31,26,32,27,28);(2,9)(3,14)(4,19)(5,6)(10,26)(11,12)(13,23)(15,18)(16,17)(20,21)(22,31)(28,32);(1,4)(2,10)(3,15)(5,20)(6,19)(7,21)(8,22)(9,25)(11,27)(12,26)(13,28)(14,29)(16,30)(17,18)(23,31)(24,32)Polycyclic presentation:
2^8|a^2=g,b^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,c^b=cg,d^b=def,f^b=fh,d^c=de,e^c=eh,f^c=fh,g^c=gh,e^d=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151236, #151238, #151240, … |
| maximal normal subgroup of | #151236, #151238, #151240, … |
| maximal quotient of | #391208, #391250, #391526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25858);