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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
647 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)(17,18,22,23)(19,20,27,24)(21,29,30,25)(26,31,32,28)(33,34,38,41)(35,39,36,43)(37,40,45,48)(42,46,44,47);(2,6)(7,12)(8,13)(14,16)(18,23)(20,24)(21,30)(26,32)(33,35,38,36)(34,39,41,43)(37,44,45,42)(40,47,48,46);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,22,27)(18,24,23,20)(21,26,30,32)(25,31,29,28)(33,36,38,35)(34,39,41,43)(37,44,45,42)(40,46,48,47);(1,3,5,9)(2,7,6,12)(4,10,11,15)(8,14,13,16)(17,20,22,24)(18,19,23,27)(21,28,30,31)(25,26,29,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,47)(45,48);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)(33,37,38,45)(34,40,41,48)(35,42,36,44)(39,46,43,47)Polycyclic presentation:
2^8|a^2=gh,b^2=fgh,c^2=g,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×#58511, #8104369, #8104434, … |
| maximal normal subgroup of | C2×#58511, #8104369, #8104434, … |
| maximal quotient of | #586493, #586676, #586677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51499);