Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3294 conjugacy classes of subgroups; only the 2839 normal subgroups are shown.
This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 2839 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) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,8,7)(3,9,12,5)(4,13,6,10)(11,15,14,16)(17,18)(19,27)(20,30)(21,24)(22,26)(23,25)(28,29)(31,32);(1,3)(2,9)(4,14)(5,7)(6,11)(8,12)(10,15)(13,16)(17,19,24,23)(18,25,21,27)(20,28,26,31)(22,29,30,32);(1,4,8,6)(2,10,7,13)(3,11,12,14)(5,15,9,16)(17,20)(18,22)(19,28)(21,30)(23,31)(24,26)(25,29)(27,32);(1,5)(2,3)(4,15)(6,16)(7,12)(8,9)(10,11)(13,14)(17,21,24,18)(19,25,23,27)(20,30,26,22)(28,29,31,32);(1,6,8,4)(2,10,7,13)(3,11,12,14)(5,16,9,15)(17,22,24,30)(18,26,21,20)(19,29,23,32)(25,31,27,28);(1,7,8,2)(3,5,12,9)(4,13,6,10)(11,15,14,16)(17,23,24,19)(18,25,21,27)(20,31,26,28)(22,29,30,32)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #63070⋊C3, #63070⋊C5 |
| maximal subgroup of | #347448, #347468, #347495, … |
| maximal normal subgroup of | #347448, #347468, #347495, … |
| maximal quotient of | #7627423, #7627538, #7627545, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56058);