Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
382 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 64 points:
(1,2,8,14)(3,15,19,33)(4,10,24,37)(5,25,28,38)(6,13,7,12)(9,20,32,18)(11,27,39,26)(16,42,44,34)(17,45,47,31)(21,50,49,30)(22,52,54,60)(23,40,29,36)(35,55,59,53)(41,48,57,46)(43,56,62,63)(51,61,64,58);(1,3,7,20,8,19,6,18)(2,9,13,33,14,32,12,15)(4,21,23,44,24,49,29,16)(5,17,27,48,28,47,26,46)(10,34,36,50,37,42,40,30)(11,31,38,57,39,45,25,41)(22,51,53,62,54,64,55,43)(35,58,52,63,59,61,60,56);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,30)(11,35)(12,36)(13,40)(14,37)(15,34)(17,43)(18,21)(19,44)(20,49)(25,52)(26,53)(27,55)(28,54)(31,56)(32,50)(33,42)(38,60)(39,59)(41,58)(45,63)(46,51)(47,62)(48,64)(57,61);(1,5,7,27,8,28,6,26)(2,11,13,38,14,39,12,25)(3,17,20,48,19,47,18,46)(4,22,29,55,24,54,23,53)(9,31,33,57,32,45,15,41)(10,35,40,60,37,59,36,52)(16,43,49,64,44,62,21,51)(30,56,42,61,50,63,34,58)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #500750, #500752, #500755, … |
| maximal normal subgroup of | #500750, #500752, #500755, … |
| maximal quotient of | #316993, #316996, #316999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21172);