Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | |
613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,8,13,12)(10,15,16,14)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28)(36,44)(39,46)(42,48)(45,47);(2,4)(7,9)(8,12)(14,15)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52818, #8092698, #8092703, … |
| maximal normal subgroup of | C2×#52818, #8092698, #8092703, … |
| maximal quotient of | #582340, #582507, #582551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45806);