Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
497 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)(5,6,10,16,12,18,11,17)(7,19,21,32,23,31,22,13)(8,14,25,33,20,34,27,36)(9,26,24,15,28,35,29,30);(5,7,12,23)(6,13,18,32)(8,24,20,29)(9,27,28,25)(10,22,11,21)(14,26,34,35)(15,33,30,36)(16,31,17,19);(2,4)(5,8,12,20)(6,14,18,34)(7,9,23,28)(10,25,11,27)(13,30,32,15)(16,33,17,36)(19,26,31,35)(21,24,22,29);(2,4)(5,9)(6,15)(7,20)(8,23)(10,29)(11,24)(12,28)(13,14)(16,26)(17,35)(18,30)(19,36)(21,25)(22,27)(31,33)(32,34)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,g^b=gh,g^c=gh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6632119, #6668416, #6668417, … |
| maximal normal subgroup of | #6632119, #6668416, #6668417, … |
| maximal quotient of | #332988, #332994, #333001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23483);