Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 |
| prime radical | #2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1078 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)(5,6,9,11)(7,10,12,8)(13,14,18,21)(15,22,20,17)(16,19,26,28)(23,27,24,25);(5,7,9,12)(6,8,11,10)(15,20)(17,22)(23,24)(25,27);(1,2)(3,4)(13,15,18,20)(14,17,21,22)(16,24,26,23)(19,27,28,25);(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,23)(17,25)(18,26)(20,24)(21,28)(22,27);(5,8,9,10)(6,7,11,12)(13,17)(14,20)(15,21)(16,25)(18,22)(19,24)(23,28)(26,27)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#44455, #7668958, #7668986, … |
| maximal normal subgroup of | C2×#44455, #7668958, #7668986, … |
| maximal quotient of | #535345, #535661, #535981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37443);