Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| 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 | |
1202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,17)(15,20,19,16);(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,20)(18,19);(1,2,3,4)(7,10)(8,12)(13,16,18,20)(14,19,17,15);(5,8,9,12)(6,10,11,7)(13,17,18,14)(15,20,19,16);(5,6)(7,12)(8,10)(9,11)Polycyclic presentation:
2^8|a^2=fh,c^2=g,d^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#35465, #7628839, #7628855, … |
| maximal normal subgroup of | C2×#35465, #7628839, #7628855, … |
| maximal quotient of | #515345, #515357, #515474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28453);