Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
681 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 681 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(2,3)(5,6)(7,11)(8,18)(9,21)(10,14)(12,20)(13,24)(15,32)(16,30)(17,27)(19,31)(22,26)(23,29)(25,34)(28,35)(33,36);(1,2)(3,4)(5,7,10,17)(6,11,14,27)(8,15,19,33)(9,22,24,16)(12,28,29,25)(13,30,21,26)(18,32,31,36)(20,35,23,34);(1,3,4,2)(5,8)(6,12)(7,15)(9,23)(10,19)(11,25)(13,31)(14,29)(16,35)(17,33)(18,21)(20,24)(22,34)(26,36)(27,28)(30,32);(8,19)(12,29)(15,33)(18,31)(20,23)(25,28)(32,36)(34,35)Polycyclic presentation:
2^8|b^2=h,c^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #394516, #394517, #394555, … |
| maximal normal subgroup of | #394516, #394517, #394555, … |
| maximal quotient of | #238564, #238567, #238571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16482);