Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1333 conjugacy classes of subgroups; only the 991 normal subgroups are shown.
This group has 1333 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)(7,16)(8,13)(9,14)(10,24)(11,15)(12,19)(17,29)(18,26)(20,30)(21,27)(22,28)(23,34)(25,33)(31,36)(32,35);(5,7,10,19,11,20,9,18)(6,12,14,16,15,26,24,30)(8,17,23,33,21,31,22,32)(13,25,28,29,27,35,34,36);(5,8)(6,13)(7,17)(9,22)(10,23)(11,21)(12,25)(14,28)(15,27)(16,29)(18,32)(19,33)(20,31)(24,34)(26,35)(30,36);(8,21)(13,27)(17,31)(22,23)(25,35)(28,34)(29,36)(32,33);(5,9,11,10)(6,14,15,24)(7,18,20,19)(8,22,21,23)(12,16,26,30)(13,28,27,34)(17,32,31,33)(25,29,35,36)Polycyclic presentation:
2^8|a^2=g,b^2=f,e^2=h,f^2=h,b^a=bf,f^a=fh,d^c=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #60108⋊C3 |
| maximal subgroup of | #146539, #146541, #146543, … |
| maximal normal subgroup of | #146539, #146541, #146543, … |
| maximal quotient of | #6344309, #6344322, #6344338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53096);