Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10,12,14)(11,13,15,16)(18,22)(23,25)(24,29)(30,31);(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(18,22)(20,28)(23,25)(26,32);(1,2)(3,7)(4,8)(5,6)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32);(1,3,5,4)(2,7,6,8)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,13,14,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #132115, #253338, #253360, … |
| maximal normal subgroup of | #132115, #253338, #253360, … |
| maximal quotient of | #515455, #515516, #515627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28473);