Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| 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×D4 | C24 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
548 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 548 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)(4,9)(5,10)(6,11)(7,12)(8,15)(14,25)(16,23)(17,22)(18,26)(19,27)(20,28)(21,29)(24,30)(31,32);(1,3,7,4)(2,8,12,22)(5,16,19,14)(6,15,20,17)(9,11,13,28)(10,24,27,21)(18,30,31,29)(23,32,25,26);(1,4,7,3)(2,9,12,13)(5,14,19,16)(6,15,20,17)(8,11,22,28)(10,25,27,23)(18,30,31,29)(21,32,24,26);(1,5,7,19)(2,10,12,27)(3,14,4,16)(6,18,20,31)(8,21,22,24)(9,23,13,25)(11,26,28,32)(15,29,17,30)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=h,b^a=be,c^a=cf,g^a=gh,d^b=dh,f^b=fgh,d^c=dh,e^c=eg,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152566, #152574, #153122, … |
| maximal normal subgroup of | #152566, #152574, #153122, … |
| maximal quotient of | #484166, #484179, #484252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26549);