Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
1793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(2,4)(6,8)(9,10)(11,16)(12,13)(14,15);(5,6)(7,8)(10,15)(11,12);(1,2,3,4)(5,6)(7,8)(9,11,14,12)(10,16,15,13);(9,12,14,11)(10,13,15,16);(9,13)(10,11)(12,15)(14,16)Polycyclic presentation:
2^8|c^2=fg,d^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #96975, #97123, #129758, … |
| maximal normal subgroup of | #96975, #97123, #129758, … |
| maximal quotient of | #516393, #516581, #516618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28156);