Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1028 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)(13,14)(15,19)(16,23)(17,18)(20,25)(21,28)(22,26)(24,27);(5,6)(7,11)(8,9)(10,12)(14,18)(19,26)(20,23)(27,28);(1,2)(3,4)(6,9)(10,11)(13,15,17,22)(14,19,18,26)(16,24,25,21)(20,28,23,27);(5,6)(7,10)(8,9)(11,12)(13,16,17,25)(14,20,18,23)(15,21,22,24)(19,27,26,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,19)(16,20)(17,18)(21,27)(22,26)(23,25)(24,28)Polycyclic presentation:
2^8|a^2=g,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#40516, #7638961, #7639570, … |
| maximal normal subgroup of | C2×#40516, #7638961, #7639570, … |
| maximal quotient of | #544447, #545029, #545389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33504);