Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#121 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1289 conjugacy classes of subgroups; only the 517 normal subgroups are shown.
This group has 1289 conjugacy classes of subgroups — too many to draw. Only its 517 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 — 5 generators on 64 points:
(1,2,8,14)(3,15,20,31)(4,21,24,38)(5,11,25,35)(6,13,7,12)(9,19,32,18)(10,28,34,23)(16,41,44,29)(17,40,45,55)(22,49,50,58)(26,37,27,36)(30,47,54,46)(33,52,57,51)(39,48,56,43)(42,60,61,53)(59,63,64,62);(1,3,7,19,8,20,6,18)(2,9,13,31,14,32,12,15)(4,16,28,48,24,44,23,43)(5,17,27,47,25,45,26,46)(10,29,38,56,34,41,21,39)(11,30,37,55,35,54,36,40)(22,42,52,63,50,61,51,62)(33,53,58,64,57,60,49,59);(1,4)(2,10)(3,16)(5,22)(6,23)(7,28)(8,24)(9,29)(11,33)(12,21)(13,38)(14,34)(15,39)(17,42)(18,43)(19,48)(20,44)(25,50)(26,51)(27,52)(30,53)(31,56)(32,41)(35,57)(36,49)(37,58)(40,59)(45,61)(46,62)(47,63)(54,60)(55,64);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,25)(9,30)(10,33)(12,36)(13,37)(14,35)(15,40)(16,42)(18,46)(19,47)(20,45)(21,49)(23,51)(24,50)(28,52)(29,53)(31,55)(32,54)(34,57)(38,58)(39,59)(41,60)(43,62)(44,61)(48,63)(56,64);(5,25)(11,35)(17,45)(22,50)(26,27)(30,54)(33,57)(36,37)(40,55)(42,61)(46,47)(49,58)(51,52)(53,60)(59,64)(62,63)Polycyclic presentation:
2^8|a^2=h,b^2=fh,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,g^c=gh,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60472⋊C3 |
| maximal subgroup of | #499906, #499907, #501081, … |
| maximal normal subgroup of | #499906, #499907, #501081, … |
| maximal quotient of | #6353805, #6353807, #6353810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53460);