Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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×#562 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
999 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)(7,13)(11,16)(14,19)(18,20)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36);(6,10)(7,13)(8,15)(18,20)(22,26)(24,33)(27,29)(30,36);(1,2)(3,4)(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36)Polycyclic presentation:
2^8|c^2=fh,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#45764, #7702392, #7702502, … |
| maximal normal subgroup of | C2×#45764, #7702392, #7702502, … |
| maximal quotient of | #557030, #557038, #557350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38752);