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 | #2673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1059 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,9)(10,12)(13,14,18,21)(15,17,22,19)(16,20,26,28)(23,25,24,27);(1,2,3,4)(15,22)(17,19)(23,24)(25,27);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,24)(17,21)(18,22)(20,25)(23,26)(27,28);(1,2)(3,4)(13,16)(14,20)(15,23)(17,25)(18,26)(19,27)(21,28)(22,24);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,15,21,22)(16,25,26,27)(20,23,28,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=gh,e^2=g,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×#44635, #7668786, #7669105, … |
| maximal normal subgroup of | C2×#44635, #7668786, #7669105, … |
| maximal quotient of | #535369, #535658, #536038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37623);