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×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
885 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,21,24)(19,25,26,32)(20,28,30,23)(22,29,31,27);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(19,26)(22,31)(25,32)(27,29);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19,21,26)(18,22,24,31)(20,29,30,27)(23,32,28,25);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16);(2,10)(7,11)(8,9)(15,16)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fh,d^2=f,g^2=h,b^a=bf,c^a=cg,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 | #245821, #245824, #245845, … |
| maximal normal subgroup of | #245821, #245824, #245845, … |
| maximal quotient of | #6386314, #6388190, #6388201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54880);