Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
5481 conjugacy classes of subgroups; only the 3135 normal subgroups are shown.
This group has 5481 conjugacy classes of subgroups — too many to draw. Only its 3135 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 — 6 generators on 36 points:
(2,4)(6,11)(9,21)(12,15)(13,22)(17,31)(20,32)(24,27)(30,36);(1,2)(3,4)(8,19)(13,22)(16,29)(20,32)(24,27)(26,33)(30,36)(34,35);(5,6,10,11)(7,15,18,12)(8,13,19,22)(9,14,21,23)(16,27,29,24)(17,28,31,25)(20,26,32,33)(30,35,36,34);(5,7,10,18)(6,12,11,15)(8,16,19,29)(9,17,21,31)(13,24,22,27)(14,25,23,28)(20,30,32,36)(26,34,33,35);(5,8)(6,13)(7,16)(9,20)(10,19)(11,22)(12,24)(14,26)(15,27)(17,30)(18,29)(21,32)(23,33)(25,34)(28,35)(31,36);(5,9)(6,14)(7,17)(8,20)(10,21)(11,23)(12,25)(13,26)(15,28)(16,30)(18,31)(19,32)(22,33)(24,34)(27,35)(29,36)Polycyclic presentation:
2^8|c^2=h,d^2=h,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #62848⋊C3 |
| maximal subgroup of | #346956, #346970, #346994, … |
| maximal normal subgroup of | #346956, #346970, #346994, … |
| maximal quotient of | #7625687, #7625688, #7625710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55836);