Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #2676 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,22)(16,19)(17,25)(18,21)(20,26)(23,27)(24,28);(5,6)(7,11)(8,12)(9,10)(13,15,18,16)(14,19,21,22)(17,23,26,24)(20,27,25,28);(1,2)(3,4)(6,10)(7,12)(13,16,18,15)(14,19,21,22)(17,24,26,23)(20,27,25,28);(5,7)(6,8)(9,12)(10,11)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,27)(25,26);(5,8)(6,7)(9,11)(10,12)(13,17,18,26)(14,20,21,25)(15,23,16,24)(19,27,22,28)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47867, #7703819, #7704091, … |
| maximal normal subgroup of | C2×#47867, #7703819, #7704091, … |
| maximal quotient of | #560499, #560535, #561599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40855);