Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C23×#6342371 (order 4096) · Inn(G) = C25 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #3531 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
814 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,26,22,24,33,38)(19,23,29,36,27,34,41,45)(20,25,31,40,32,37,44,47)(28,35,42,46,30,39,43,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,30)(21,29)(22,27)(24,34)(25,39)(26,36)(28,32)(31,43)(33,41)(35,37)(38,45)(40,48)(42,44)(46,47);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(19,27)(25,37)(26,38)(28,30)(29,41)(40,47)(42,43);(2,8)(6,9)(7,16)(14,15)(19,27)(23,34)(28,30)(29,41)(35,39)(36,45)(42,43)(46,48);(17,20,21,31,22,32,33,44)(18,25,26,40,24,37,38,47)(19,28,29,42,27,30,41,43)(23,35,36,46,34,39,45,48)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=df,c^b=cfh,d^b=dh,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129678, #210398, #210399, … |
| maximal normal subgroup of | #129678, #210398, #210399, … |
| maximal quotient of | #6370584, #6370593, #6370595, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53334);