Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
293 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 61 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 — 3 generators on 36 points:
(5,6)(7,18)(8,12)(9,25)(10,28)(11,17)(13,20)(14,22)(15,26)(16,24)(19,31)(21,30)(23,32)(27,35)(29,34)(33,36);(1,2)(3,4)(6,12)(7,19)(8,22)(9,26)(10,24)(13,31)(14,16)(15,28)(17,25)(18,35)(20,36)(21,29)(30,32)(33,34);(2,4)(5,7)(6,13)(8,23)(9,27)(10,29)(11,21)(12,30)(14,31)(15,34)(16,35)(17,33)(18,25)(19,24)(20,26)(22,36)(28,32)Polycyclic presentation:
2^8|d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dgh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13336, #498877, #498878, … |
| maximal normal subgroup of | C2×#13336, #498877, #498878, … |
| maximal quotient of | #142632, #142633, #142644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6324);