Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
336 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 119 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,22,26,30,25,29,34,36)(23,31,32,28,24,27,33,35);(1,2)(3,4)(6,10)(11,17)(12,18)(19,20)(21,23)(22,27)(24,25)(26,32)(28,36)(29,31)(30,35)(33,34);(2,4)(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,24,25,23)(22,28,29,35)(26,32,34,33)(27,30,31,36);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(23,24)(27,31)(28,35)(32,33)Polycyclic presentation:
2^8|a^2=f,c^2=e,d^2=eh,f^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #488660, #488661, C2×#29882, … |
| maximal normal subgroup of | #488660, #488661, C2×#29882, … |
| maximal quotient of | #330775, #330778, #330781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22870);