Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | |
1084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,10)(8,12)(13,14)(15,21)(16,23)(17,25)(18,19)(20,26)(22,24)(27,28);(6,10)(7,11)(14,19)(16,21)(20,25)(24,27);(1,2,3,4)(5,6,9,10)(7,8,11,12)(13,15)(14,16)(17,22)(18,23)(19,21)(20,24)(25,27)(26,28);(5,7,9,11)(6,8,10,12)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,28)(26,27);(5,8)(6,11)(7,10)(9,12)(13,17,18,26)(14,20,19,25)(15,22,23,28)(16,24,21,27)Polycyclic presentation:
2^8|c^2=fg,d^2=h,e^2=f,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 | C2×#37133, #7634501, #7634504, … |
| maximal normal subgroup of | C2×#37133, #7634501, #7634504, … |
| maximal quotient of | #553776, #553781, #554031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30121);