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 | |
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:
(1,2,4,3)(5,6)(7,8)(9,10)(11,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25);(6,10)(7,11)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27);(1,2,4,3)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28);(5,7,9,11)(6,8,10,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(1,3,4,2)(5,8)(6,11)(7,10)(9,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=gh,d^2=f,e^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×#34525, #7626155, #7626158, … |
| maximal normal subgroup of | C2×#34525, #7626155, #7626158, … |
| maximal quotient of | #513874, #513881, #513950, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27513);