Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,19,17)(15,22,16,20)(18,25,26,21)(23,27,24,28);(5,6)(7,8)(9,10)(11,12)(13,15,19,16)(14,20,17,22)(18,23,26,24)(21,27,25,28);(1,2)(3,4)(6,10)(7,11)(13,16,19,15)(14,20,17,22)(18,24,26,23)(21,27,25,28);(5,7,9,11)(6,8,10,12)(13,17,19,14)(15,20,16,22)(18,25,26,21)(23,27,24,28);(5,8)(6,11)(7,10)(9,12)(13,18,19,26)(14,21,17,25)(15,23,16,24)(20,27,22,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=f,d^2=fh,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47688, #7701706, #7701707, … |
| maximal normal subgroup of | C2×#47688, #7701706, #7701707, … |
| maximal quotient of | #576634, #576697, #576730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40676);