Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | Aut(G) = #72256403 (order 2048) · Inn(G) = #1790 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | #3545 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
389 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 389 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2)(3,11)(4,15,17,8)(5,18,19,9)(6,10)(7,13,21,28)(12,24,27,31)(14,25)(16,22)(20,26)(23,30)(29,32);(1,3,6,14)(2,7,10,21)(4,12,17,27)(5,13,19,28)(8,20,15,29)(9,11,18,25)(16,26,30,32)(22,31,23,24);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,23)(10,15)(11,24)(13,26)(14,27)(18,22)(19,30)(21,29)(25,31)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=egh,f^b=fg,d^c=dg,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152569, #152592, #152593, … |
| maximal normal subgroup of | #152569, #152592, #152593, … |
| maximal quotient of | #129576, #129578, #129580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5756);