Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C4×#121 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,23)(3,14,5,11,17,28,16,9)(4,18,21,10,22,26,20,27)(15,24,19,29,31,25,30,32);(1,3,6,16,8,17,7,5)(2,9,12,14,13,11,23,28)(4,15,20,30,22,31,21,19)(10,24,26,29,27,25,18,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,22)(9,24)(11,25)(12,26)(13,27)(14,29)(16,30)(17,31)(18,23)(28,32);(1,5,7,17,8,16,6,3)(2,11,12,28,13,9,23,14)(4,19,21,31,22,30,20,15)(10,25,26,32,27,24,18,29)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #136598, #136601, #136627, … |
| maximal normal subgroup of | #136598, #136601, #136627, … |
| maximal quotient of | C8×#471, C8×#472, C8×#473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14546);