Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22)(9,16)(10,15)(11,20)(12,24)(13,23)(14,27)(18,33)(19,28)(21,30)(25,32)(26,29)(31,35)(34,36);(5,7,10,21)(6,11,15,29)(8,19,24,34)(9,26,27,20)(12,31,22,33)(13,28,32,36)(14,30,16,17)(18,23,35,25);(2,4)(5,8)(6,12)(7,18)(9,23)(10,24)(11,28)(13,14)(15,22)(16,32)(17,31)(19,26)(20,34)(21,35)(25,27)(29,36)(30,33);(5,8)(6,13)(7,19)(9,25)(10,24)(11,28)(12,14)(15,32)(16,22)(17,33)(18,26)(20,35)(21,34)(23,27)(29,36)(30,31)Polycyclic presentation:
2^8|b^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,g^c=gh,e^d=eh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23889, #6437601, #6437604, … |
| maximal normal subgroup of | C2×#23889, #6437601, #6437604, … |
| maximal quotient of | #244501, #244503, #244507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16877);