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 | |
1103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(7,11)(9,12)(14,18)(19,20);(2,4)(5,6)(7,12)(8,10)(9,11);(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,19,16,20);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,16)(14,19,18,20);(5,7)(6,9)(8,11)(10,12)(13,16,17,15)(14,20,18,19)Polycyclic presentation:
2^8|c^2=gh,d^2=g,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×#44382, #7668148, #7668728, … |
| maximal normal subgroup of | C2×#44382, #7668148, #7668728, … |
| maximal quotient of | #535349, #535658, #536001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37370);