Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
1167 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)(13,14,18,21)(15,16,23,19)(17,20,25,27)(22,24,28,26);(5,6)(7,11)(8,9)(10,12)(13,15,18,23)(14,19,21,16)(17,22,25,28)(20,26,27,24);(1,2,3,4)(6,9)(10,11)(13,15)(14,16)(17,22)(18,23)(19,21)(20,24)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,24)(18,19)(20,22)(21,23)(25,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,17,18,25)(14,20,21,27)(15,22,23,28)(16,24,19,26)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#40599, #7639278, #7639305, … |
| maximal normal subgroup of | C2×#40599, #7639278, #7639305, … |
| maximal quotient of | #544530, #545089, #545528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33587);