Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,11)(8,18)(9,21)(10,13)(12,19)(14,23)(15,29)(16,25)(17,33)(20,28)(22,27)(24,31)(26,32)(30,36)(34,35);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,25)(14,26)(18,29)(19,31)(20,30)(21,33)(22,34)(23,32)(27,35)(28,36);(5,8,10,20)(6,12,13,27)(7,15,16,30)(9,22,23,19)(11,24,25,35)(14,28,21,18)(17,34,32,31)(26,36,33,29);(7,16)(11,25)(15,30)(17,32)(24,35)(26,33)(29,36)(31,34);(6,13)(11,25)(12,27)(14,21)(18,28)(24,35)(26,33)(29,36)Polycyclic presentation:
2^8|b^2=f,c^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #247692, #247694, #247716, … |
| maximal normal subgroup of | #247692, #247694, #247716, … |
| maximal quotient of | #6355305, #6355306, #6355307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53412);