Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(19,25)(23,28)(26,31)(30,32)(33,34)(35,38)(36,41)(37,42)(39,43)(40,45)(44,46)(47,48);(2,6)(7,12)(8,13)(14,16)(18,22)(20,27)(23,28)(26,31)(34,35)(36,43)(40,42)(44,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)Polycyclic presentation:
2^8|c^2=fgh,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#53384, #8105932, #8105934, … |
| maximal normal subgroup of | C2×#53384, #8105932, #8105934, … |
| maximal quotient of | #606851, #606853, #606854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46372);