Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| 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 | |
1005 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)(5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,17)(18,26)(19,21)(20,27)(23,25)(24,28);(5,7,8,12)(6,9,10,11)(13,15,19,16)(14,17,21,22)(18,23,27,24)(20,25,26,28);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,15)(14,17,21,22)(18,24,27,23)(20,25,26,28);(5,6)(7,9)(8,10)(11,12)(13,17)(14,15)(16,21)(18,25)(19,22)(20,23)(24,26)(27,28);(13,18,19,27)(14,20,21,26)(15,23,16,24)(17,25,22,28)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #214051, #214058, #6481814, … |
| maximal normal subgroup of | #214051, #214058, #6481814, … |
| maximal quotient of | #553753, #553754, #554017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30377);