Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(21,22,27,32)(23,29,24,28)(25,26,31,30)(33,34,35,36);(5,6,9,12)(7,10,15,13)(8,16,17,11)(14,19,20,18)(21,23,27,24)(22,28,32,29)(25,33,31,35)(26,36,30,34);(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,17)(18,20)(21,24,27,23)(22,29,32,28)(25,33,31,35)(26,36,30,34);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,25,27,31)(22,30,32,26)(23,33,24,35)(28,34,29,36);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,26,27,30)(22,31,32,25)(23,34,24,36)(28,33,29,35)Polycyclic presentation:
2^8|a^2=f,b^2=fh,c^2=g,d^2=f,e^2=fh,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 | #132372, #248401, #248402, … |
| maximal normal subgroup of | #132372, #248401, #248402, … |
| maximal quotient of | #544521, #545078, #545510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35799);