Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(4,8)(6,7)(11,15)(13,16)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,29)(31,32);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(25,28)(27,32);(1,2,5,3)(4,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29);(1,3,5,2)(4,7,8,6)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,4)(2,6)(3,7)(5,8)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)Polycyclic presentation:
2^8|b^2=g,c^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#37237, #7634552, #7634565, … |
| maximal normal subgroup of | C2×#37237, #7634552, #7634565, … |
| maximal quotient of | #553741, #553749, #554032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30225);