Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) = C24×D4×#120 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,18,13,27)(11,16,23,30,21,31,22,37)(12,24,25,17,26,33,28,32)(20,34,36,29,35,39,38,40);(2,5)(6,8)(9,11,14,23,15,21,13,22)(10,16,19,30,18,31,27,37)(12,20,25,36,26,35,28,38)(17,29,33,39,32,40,24,34);(9,12)(10,17)(11,20)(13,28)(14,25)(15,26)(16,29)(18,32)(19,33)(21,35)(22,38)(23,36)(24,27)(30,39)(31,40)(34,37);(1,3,4,7)(2,6,5,8)(10,18)(11,21)(17,32)(19,27)(20,35)(22,23)(24,33)(36,38)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #222816, #222817, #222818, … |
| maximal normal subgroup of | #222816, #222817, #222818, … |
| maximal quotient of | #203192, #203193, #203194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14128);