Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | Aut(G) = #72342352 (order 4096) · Inn(G) = C2×B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C24 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
542 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 109 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)(3,13)(4,10)(5,19)(6,9)(7,12)(8,15)(11,22)(14,25)(16,29)(17,28)(18,26)(20,27)(21,30)(23,32)(24,31);(1,3,7,17)(2,8,12,25)(4,15,5,14)(6,21,22,16)(9,23,11,24)(10,28,19,13)(18,31,20,32)(26,30,27,29);(2,9)(6,22)(8,23)(10,26)(11,12)(13,29)(16,21)(18,20)(19,27)(24,25)(28,30)(31,32);(1,4,7,5)(2,10,12,19)(3,14,17,15)(6,18,22,20)(8,13,25,28)(9,26,11,27)(16,31,21,32)(23,29,24,30)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,d^b=dh,e^b=eh,f^b=fgh,e^c=eg,f^c=fh,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152563, #152571, #153128, … |
| maximal normal subgroup of | #152563, #152571, #153128, … |
| maximal quotient of | #484179, #484185, #484265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26559);