Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11912 |
| automorphism | Aut(G) = #72306341 (order 2048) · Inn(G) = #1530 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1530 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #97 | C23 |
| prime radical | C23 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,6,12,26,42,7,13,25,43,28,10,37,39)(3,14,16,45,17,8,29,31,18,32,50,40,27,33,23,56)(4,19,21,51,22,36,54,41,20,38,24,58,52,9,34,35)(15,30,47,60,48,59,57,63,46,61,49,62,55,44,53,64);(1,3,7,18)(2,8,13,33)(4,15,20,46)(5,23,25,29)(6,27,28,17)(9,30,36,61)(10,32,12,14)(11,40,43,45)(16,26,50,37)(19,44,38,59)(21,53,24,57)(22,55,52,48)(31,39,56,42)(34,47,54,49)(35,63,41,64)(51,60,58,62);(1,4,7,20)(2,9)(3,15,18,46)(5,24,25,21)(6,22,28,52)(8,30)(10,38)(11,41)(12,19)(13,36)(14,44)(16,49,50,47)(17,48,27,55)(23,57,29,53)(26,34,37,54)(31,62)(32,59)(33,61)(35,43)(39,51)(40,64)(42,58)(45,63)(56,60)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=gh,d^2=g,e^2=f,f^2=h,b^a=bd,c^a=cf,d^a=df,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,d^c=dh,e^c=eh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #132070, #132071, #132073, … |
| maximal normal subgroup of | #132070, #132071, #132073, … |
| maximal quotient of | #124338, #124459, #124496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4902);