Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #62636 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×D8 |
| commutator, Frattini, cosocle | C8 | C25 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1039 conjugacy classes of subgroups; only the 477 normal subgroups are shown.
This group has 1039 conjugacy classes of subgroups — too many to draw. Only its 477 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 — 5 generators on 32 points:
(1,2)(3,8)(4,7)(5,18)(6,9)(10,15)(11,16)(12,22)(13,19)(14,28)(17,20)(21,29)(23,27)(24,26)(25,30)(31,32);(2,7)(4,15)(5,16)(8,19)(9,20)(10,11)(13,27)(14,26)(17,29)(18,21)(22,30)(23,24)(25,32)(28,31);(1,3)(2,8)(4,13)(5,14)(6,12)(7,19)(9,22)(10,23)(11,24)(15,27)(16,26)(17,25)(18,28)(20,30)(21,31)(29,32);(3,12)(8,22)(13,25)(14,26)(19,30)(23,31)(24,28)(27,32);(2,9)(7,20)(8,22)(10,21)(11,18)(19,30)(23,31)(24,28)Polycyclic presentation:
2^8|f^2=gh,g^2=h,b^a=bf,e^a=eh,f^a=fg,g^a=gh,f^b=fg,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #62636⋊C3 |
| maximal subgroup of | #497430, #499742, #499743, … |
| maximal normal subgroup of | #497430, #499742, #499743, … |
| maximal quotient of | #7622497, #7622499, #7622537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55624);