Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
611 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,18,22,21)(19,25,26,23)(20,28,29,24)(27,30,32,31);(2,6)(3,9)(4,13)(14,16)(19,26)(23,25)(27,32)(30,31)(33,34,38,36)(35,41,43,39)(37,44,45,40)(42,46,48,47);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,19,22,26)(18,23,21,25)(20,27,29,32)(24,30,28,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20,22,29)(18,24,21,28)(19,27,26,32)(23,30,25,31)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,18)(19,25,26,23)(20,28,29,24)(27,31,32,30)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=h,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58765, #8105006, #8105009, … |
| maximal normal subgroup of | C2×#58765, #8105006, #8105009, … |
| maximal quotient of | #586557, #586613, #586677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51753);