Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,28,22,27,23,26)(19,29,31,25,20,24,30,32);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,21,31,22,20,23,30)(18,24,28,32,27,29,26,25);(10,14)(13,16)(17,20,22,19)(18,25,27,32)(21,31,23,30)(24,28,29,26);(1,4,5,3)(2,7,8,6)(17,21,22,23)(18,26,27,28)(19,30,20,31)(24,32,29,25)Polycyclic presentation:
2^8|a^2=eg,b^2=efh,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27831, #6458336, #6458342, … |
| maximal normal subgroup of | C2×#27831, #6458336, #6458342, … |
| maximal quotient of | #360459, #363093, #366462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20819);