Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | Aut(G) = #72339816 (order 8192) · Inn(G) = #555 · Out(G) = #1600 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2571 |
| prime radical | C23 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,13)(4,18,19,23)(5,8,6,11)(9,14)(10,21,26,20)(15,29)(16,24)(17,22)(25,31)(27,28)(30,32);(2,8,12,11)(3,14,17,16)(4,19)(9,24)(10,18,26,23)(13,22)(15,30,31,28)(20,21);(1,3)(2,9,12,24)(4,15)(5,14)(6,16)(7,17)(8,22,11,13)(10,25,26,29)(18,27,23,32)(19,31)(20,28)(21,30);(1,4,5,20,7,19,6,21)(2,10,11,18,12,26,8,23)(3,15,14,28,17,31,16,30)(9,25,13,27,24,29,22,32)Polycyclic presentation:
2^8|a^2=fg,b^2=eg,c^2=ef,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #151316, #151317, #510293, … |
| maximal normal subgroup of | #151316, #151317, #510293, … |
| maximal quotient of | #362305, #365888, #369552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22219);