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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1050 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)(16,24)(17,20)(18,19)(21,26)(23,27)(25,28);(6,9)(7,11)(13,15,19,17)(14,20,18,22)(16,25,26,23)(21,28,24,27);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,16,19,26)(14,21,18,24)(15,23,17,25)(20,27,22,28);(5,6)(7,10)(8,9)(11,12)(13,17,19,15)(14,20,18,22)(16,23,26,25)(21,28,24,27);(5,7)(6,10)(8,11)(9,12)(13,18,19,14)(15,20,17,22)(16,24,26,21)(23,27,25,28)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=h,d^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×#49584, #7705619, #7705663, … |
| maximal normal subgroup of | C2×#49584, #7705619, #7705663, … |
| maximal quotient of | #560502, #560527, #561594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 42572);