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 | |
1009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,14)(7,11,15,20)(8,12,18,17)(9,13,19,16)(22,23)(24,31)(28,30)(32,36);(7,15)(9,19)(11,20)(13,16)(21,22,26,23)(24,27,31,29)(25,30,33,28)(32,35,36,34);(1,2,3,4)(5,7)(6,11)(8,13)(9,17)(10,15)(12,19)(14,20)(16,18)(21,23,26,22)(24,27,31,29)(25,30,33,28)(32,34,36,35);(5,8)(6,12)(7,16)(9,11)(10,18)(13,15)(14,17)(19,20)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,9)(6,13)(7,17)(8,11)(10,19)(12,15)(14,16)(18,20)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,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×#41416, #7639992, #7640048, … |
| maximal normal subgroup of | C2×#41416, #7639992, #7640048, … |
| maximal quotient of | #544531, #545088, #545529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34404);