Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2431 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,32)(28,30)(29,31);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,22)(24,31)(28,29)(30,32);(10,14)(13,16)(17,19,21,28)(18,24,23,30)(20,27,22,29)(25,32,26,31);(2,6)(3,8)(10,14)(13,16)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,32,30,31)Polycyclic presentation:
2^8|a^2=eh,b^2=efh,c^2=h,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6632149, #6632151, #6632181, … |
| maximal normal subgroup of | #6632149, #6632151, #6632181, … |
| maximal quotient of | #383254, #383876, #383910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24305);