Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
257 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 257 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,8,14)(3,15,20,48)(4,21,25,45)(5,26,31,43)(6,13,7,12)(9,33,37,17)(10,32,40,24)(11,29,44,30)(16,49,52,64)(18,47,19,41)(22,56,51,46)(23,42,28,39)(27,36,54,35)(34,60,63,57)(38,53,62,50)(55,61,59,58);(1,3)(2,9)(4,22)(5,27)(6,18)(7,19)(8,20)(10,38)(11,41)(12,35)(13,36)(14,37)(15,26)(16,24)(17,30)(21,34)(23,53)(25,51)(28,50)(29,33)(31,54)(32,52)(39,46)(40,62)(42,56)(43,48)(44,47)(45,63)(49,55)(57,58)(59,64)(60,61);(1,4)(2,10)(3,16)(5,28)(6,32)(7,24)(8,25)(9,34)(11,42)(12,45)(13,21)(14,40)(15,46)(17,53)(18,22)(19,51)(20,52)(23,31)(26,59)(27,60)(29,58)(30,61)(33,50)(35,38)(36,62)(37,63)(39,44)(41,64)(43,55)(47,49)(48,56)(54,57)Polycyclic presentation:
2^8|a^2=h,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,f^b=fh,g^b=gh,d^c=dh,e^c=eh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13324, #498837, #499515, … |
| maximal normal subgroup of | C2×#13324, #498837, #499515, … |
| maximal quotient of | #142247, #142253, #142255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6312);