Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×2−1+4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2405 conjugacy classes of subgroups; only the 991 normal subgroups are shown.
This group has 2405 conjugacy classes of subgroups — too many to draw. Only its 991 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 — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,15)(8,12)(9,21)(10,13)(11,17)(14,23)(16,29)(18,28)(19,26)(20,33)(22,25)(24,31)(27,32)(30,36)(34,35);(5,7,10,18)(6,11,13,25)(8,16,19,30)(9,22,23,17)(12,24,26,35)(14,28,21,15)(20,34,32,31)(27,36,33,29);(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,26)(14,27)(15,29)(17,31)(18,30)(21,33)(22,34)(23,32)(25,35)(28,36);(8,19)(12,26)(16,30)(20,32)(24,35)(27,33)(29,36)(31,34);(6,13)(11,25)(12,26)(14,21)(15,28)(24,35)(27,33)(29,36)Polycyclic presentation:
2^8|a^2=g,b^2=h,f^2=h,b^a=bf,e^a=eh,f^a=fh,f^b=fh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #60088⋊C3 |
| maximal subgroup of | #359646, #359648, #359651, … |
| maximal normal subgroup of | #359646, #359648, #359651, … |
| maximal quotient of | #6342909, #6342929, #6342935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53076);