Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11989 |
| automorphism | Aut(G) = #72306341 (order 2048) · Inn(G) = #1551 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1551 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C23 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,11,6,10,24,36,7,13,25,39,27,12,40,38)(3,14,16,42,17,31,46,56,18,30,47,37,26,8,23,29)(4,19,21,48,22,9,32,33,20,35,49,61,50,34,58,60)(15,28,44,54,45,41,62,63,43,55,57,64,52,53,51,59);(1,3,7,18)(2,8,13,31)(4,15,20,43)(5,23,25,46)(6,26,27,17)(9,28,34,55)(10,30,12,14)(11,37,39,42)(16,24,47,40)(19,41,35,53)(21,51,49,62)(22,52,50,45)(29,36,56,38)(32,57,58,44)(33,59,60,63)(48,54,61,64);(1,4)(2,9,13,34)(3,15)(5,21)(6,22)(7,20)(8,28,31,55)(10,35,12,19)(11,33,39,60)(14,41,30,53)(16,44)(17,45)(18,43)(23,51)(24,32)(25,49)(26,52)(27,50)(29,54,56,64)(36,61,38,48)(37,59,42,63)(40,58)(46,62)(47,57)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=g,d^2=g,e^2=fg,f^2=h,b^a=bd,c^a=cf,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | #132070, #132071, #132073, … |
| maximal normal subgroup of | #132070, #132071, #132073, … |
| maximal quotient of | #124351, #124411, #124719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4978);