Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33388 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#102 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
564 conjugacy classes of subgroups; only the 221 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,9,21)(4,14)(5,17)(8,20,23,31)(10,15,18,22)(12,27)(13,26)(16,19)(24,28,29,32)(25,30);(1,2)(3,8)(4,15)(5,18)(6,9)(7,19)(10,17)(11,23)(12,28)(13,29)(14,21)(16,22)(20,30)(24,26)(25,32)(27,31);(1,3)(2,8)(4,12)(5,13)(6,11)(7,20)(9,23)(10,24)(14,27)(15,28)(16,25)(17,26)(18,29)(19,30)(21,31)(22,32);(3,11)(8,23)(12,25)(13,26)(20,31)(24,29)(27,30)(28,32)Polycyclic presentation:
2^8|a^2=f,e^2=g,g^2=h,b^a=be,e^a=eg,g^a=gh,e^b=egh,f^b=fh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #33388⋊C3 |
| maximal subgroup of | #497408, #497409, #498126, … |
| maximal normal subgroup of | #497408, #497409, #498126, … |
| maximal quotient of | #481066, #481067, #481284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26376);