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, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28)(33,34,38,41)(35,39,36,43)(37,40,45,48)(42,46,44,47);(2,6)(3,9)(4,13)(14,16)(18,23)(20,27)(25,29)(28,32)(33,35,38,36)(34,39,41,43)(37,44,45,42)(40,47,48,46);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32)(33,36,38,35)(34,39,41,43)(37,44,45,42)(40,46,48,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,47)(45,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,37,38,45)(34,40,41,48)(35,42,36,44)(39,46,43,47)Polycyclic presentation:
2^8|a^2=fgh,b^2=g,c^2=fg,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×#54576, #8103866, #8103904, … |
| maximal normal subgroup of | C2×#54576, #8103866, #8103904, … |
| maximal quotient of | #586276, #586629, #587050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47564);