Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| 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 | |
963 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)(6,10)(8,15)(11,16)(14,19)(21,22,26,27)(23,24,31,28)(25,33,34,29)(30,35,36,32);(6,10)(7,13)(12,17)(14,19)(22,27)(24,28)(29,33)(32,35);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23)(22,28)(24,27)(25,30)(26,31)(29,35)(32,33)(34,36);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,28)(22,23,27,31)(25,32,34,35)(29,30,33,36);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,34)(22,29,27,33)(23,30,31,36)(24,32,28,35)Polycyclic presentation:
2^8|a^2=f,c^2=fgh,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #214049, #214655, #6481601, … |
| maximal normal subgroup of | #214049, #214655, #6481601, … |
| maximal quotient of | #553765, #553771, #554023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30411);