Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12465 |
| automorphism | Aut(G) = #72309547 (order 1024) · Inn(G) = #1677 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1677 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,9,16)(3,17,24,49)(4,25,29,41)(5,13,30,35)(6,12,18,44)(7,15,8,14)(10,34,40,20)(11,26,43,27)(19,51,54,63)(21,38,31,39)(22,37,23,45)(28,58,33,46)(32,47,50,42)(36,57,61,53)(48,55,60,52)(56,62,59,64);(1,3,18,21,7,22,5,20,9,24,6,31,8,23,30,34)(2,10,35,37,14,38,12,17,16,40,13,45,15,39,44,49)(4,19,50,56,28,53,27,55,29,54,32,59,33,57,26,52)(11,36,58,62,42,51,25,48,43,61,46,64,47,63,41,60);(1,4)(2,11)(3,19)(5,27)(6,32)(7,28)(8,33)(9,29)(10,36)(12,25)(13,46)(14,42)(15,47)(16,43)(17,48)(18,50)(20,55)(21,56)(22,53)(23,57)(24,54)(26,30)(31,59)(34,52)(35,58)(37,62)(38,51)(39,63)(40,61)(41,44)(45,64)(49,60)Polycyclic presentation:
2^8|a^2=h,b^2=dg,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,e^b=eh,d^c=dh,e^c=eg,g^c=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12466, #490558, #490560, … |
| maximal normal subgroup of | C2×#12466, #490558, #490560, … |
| maximal quotient of | #134990, #138573, #138668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5454);