Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | Aut(G) order 16384 |
| 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 | C2×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
491 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 491 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(8,14,25,33,20,30,27,34)(9,26,24,35,28,36,29,15);(5,7,12,23)(6,13,18,32)(8,24,20,29)(9,27,28,25)(10,22,11,21)(14,26,30,36)(15,34,35,33)(16,31,17,19);(2,4)(5,8,12,20)(6,14,18,30)(7,9,23,28)(10,25,11,27)(13,15,32,35)(16,33,17,34)(19,26,31,36)(21,24,22,29);(2,4)(5,9)(6,15)(7,20)(8,23)(10,29)(11,24)(12,28)(13,30)(14,32)(16,36)(17,26)(18,35)(19,33)(21,25)(22,27)(31,34)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=def,c^b=cg,d^b=de,e^b=eh,g^b=gh,g^c=gh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #153141, #153142, #153146, … |
| maximal normal subgroup of | #153141, #153142, #153146, … |
| maximal quotient of | #332843, #332849, #332857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23476);