Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13459 |
| automorphism | Aut(G) = #72317053 (order 4096) · Inn(G) = #498 · Out(G) = #392 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #498 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2084 |
| prime radical | C23 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,7,5,10,18,22)(3,11,14,27,13,8,15,24)(4,16,19,20,21,25,23,9)(12,17,29,26,31,32,30,28);(1,3,5,13)(2,8,10,11)(4,17)(6,14,18,15)(7,24,22,27)(9,12)(16,29)(19,26)(20,31)(21,32)(23,28)(25,30);(1,4,18,19,5,21,6,23)(2,9,22,16,10,20,7,25)(3,12,14,30,13,31,15,29)(8,26,24,17,11,28,27,32)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=dg,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,f^b=fg,g^b=gh,d^c=dh,e^c=eh,f^c=fg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #126349, #126350, #126437, … |
| maximal normal subgroup of | #126349, #126350, #126437, … |
| maximal quotient of | #146153, #146158, #146168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6499);