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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
923 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)(5,6,10,11)(7,8,15,12)(9,17,18,13)(14,19,20,16)(21,22,26,29)(23,24,27,32)(25,33,34,28)(30,36,35,31);(6,11)(8,12)(13,17)(16,19)(21,23)(22,24)(25,30)(26,27)(28,31)(29,32)(33,36)(34,35);(1,2,3,4)(5,7)(6,12)(8,11)(9,14)(10,15)(13,19)(16,17)(18,20)(21,24)(22,27)(23,29)(25,31)(26,32)(28,35)(30,33)(34,36);(5,8,10,12)(6,7,11,15)(9,16,18,19)(13,14,17,20)(21,24)(22,27)(23,29)(25,31)(26,32)(28,35)(30,33)(34,36);(5,9,10,18)(6,13,11,17)(7,14,15,20)(8,16,12,19)(21,25,26,34)(22,28,29,33)(23,30,27,35)(24,31,32,36)Polycyclic presentation:
2^8|a^2=fh,c^2=fg,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 | #214047, #214653, #6481603, … |
| maximal normal subgroup of | #214047, #214653, #6481603, … |
| maximal quotient of | #553766, #553769, #554022, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30380);