Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
995 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(6,10)(7,13)(12,17)(14,19)(21,22,25,26)(23,29,30,27)(24,32,33,28)(31,34,36,35);(6,10)(11,16)(12,17)(18,20)(22,26)(23,30)(28,32)(31,36);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,22)(23,27)(24,28)(25,26)(29,30)(31,34)(32,33)(35,36);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23,25,30)(22,27,26,29)(24,31,33,36)(28,34,32,35);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24,25,33)(22,28,26,32)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^8|a^2=h,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #107484, #108377, #190387, … |
| maximal normal subgroup of | #107484, #108377, #190387, … |
| maximal quotient of | #513792, #513810, #513818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27383);