Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(13,14,18,16)(15,19,23,21)(17,24,25,20)(22,27,28,26);(5,6,8,9)(7,10,11,12)(14,16)(19,21)(20,24)(26,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,27,28,26);(5,7)(6,10)(8,11)(9,12)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fg,d^2=h,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#34486, #7626106, #7626133, … |
| maximal normal subgroup of | C2×#34486, #7626106, #7626133, … |
| maximal quotient of | #513828, #513844, #513855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27474);