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 | |
3192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,7,6)(3,10,13,8)(4,14,5,9)(11,15,12,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31);(1,3,7,13)(2,8,6,10)(4,11,5,12)(9,15,14,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,4,7,5)(2,9,6,14)(3,11,13,12)(8,15,10,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31);(1,3)(2,8)(4,11)(5,12)(6,10)(7,13)(9,15)(14,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32);(1,5,7,4)(2,9,6,14)(3,12,13,11)(8,15,10,16);(1,6,7,2)(3,10,13,8)(4,14,5,9)(11,16,12,15)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^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 | #63061⋊C3 |
| maximal subgroup of | #347451, #347471, #347499, … |
| maximal normal subgroup of | #347451, #347471, #347499, … |
| maximal quotient of | #7627416, #7627449, #7627454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56049);