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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(14,15)(16,23)(20,22)(24,28);(5,6)(7,11)(8,9)(10,12)(13,14,18,15)(16,19,23,21)(17,22,25,20)(24,27,28,26);(1,2)(3,4)(6,9)(10,11)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,26,28,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)Polycyclic presentation:
2^8|b^2=f,c^2=f,e^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×#40499, #7639305, #7639330, … |
| maximal normal subgroup of | C2×#40499, #7639305, #7639330, … |
| maximal quotient of | #544466, #545045, #545396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33487);