Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61569 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#555 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
899 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,26)(19,27,28,32)(20,25,21,24)(23,30,31,29);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(19,28)(23,31)(27,32)(29,30);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,32)(26,31);(3,11)(6,9)(10,16)(13,15)(17,20,22,21)(18,24,26,25)(19,29,28,30)(23,27,31,32);(2,8)(6,9)(7,14)(13,15)Polycyclic presentation:
2^8|a^2=h,c^2=f,d^2=h,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gh,c^b=ch,d^b=df,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #245628, #245631, #245652, … |
| maximal normal subgroup of | #245628, #245631, #245652, … |
| maximal quotient of | #6386154, #6387813, #6388188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54709);