Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1557 conjugacy classes of subgroups; only the 563 normal subgroups are shown.
This group has 1557 conjugacy classes of subgroups — too many to draw. Only its 563 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 — 5 generators on 36 points:
(2,4)(5,6,10,14)(7,15,19,29)(8,12,20,26)(9,22,16,13)(11,23,25,18)(17,28,31,36)(21,33,30,27)(24,34,35,32);(5,7,16,23,10,19,9,18)(6,11,22,29,14,25,13,15)(8,17,30,34,20,31,21,32)(12,24,33,36,26,35,27,28);(1,2,3,4);(5,8)(6,12)(7,17)(9,21)(10,20)(11,24)(13,27)(14,26)(15,28)(16,30)(18,32)(19,31)(22,33)(23,34)(25,35)(29,36);(8,20)(12,26)(17,31)(21,30)(24,35)(27,33)(28,36)(32,34)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=g,f^2=h,b^a=bf,c^a=cg,f^a=fh,e^d=eh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #60648⋊C3 |
| maximal subgroup of | #130000, #130004, #152055, … |
| maximal normal subgroup of | #130000, #130004, #152055, … |
| maximal quotient of | #6357191, #6357195, #6357197, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53636);