Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | C2×#538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1060 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)(13,14,18,22)(15,16,20,19)(17,25,27,21)(23,28,26,24);(5,6)(7,8)(9,10)(11,12)(13,15,18,20)(14,19,22,16)(17,26,27,23)(21,24,25,28);(1,2)(3,4)(6,10)(7,11)(13,15)(14,16)(17,23)(18,20)(19,22)(21,24)(25,28)(26,27);(5,7,9,11)(6,8,10,12)(13,16)(14,20)(15,22)(17,24)(18,19)(21,26)(23,25)(27,28);(5,8)(6,11)(7,10)(9,12)(13,17,18,27)(14,21,22,25)(15,23,20,26)(16,24,19,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=fh,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47684, #7701697, #7701707, … |
| maximal normal subgroup of | C2×#47684, #7701697, #7701707, … |
| maximal quotient of | #576634, #576696, #576731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40672);