Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C25 |
| socle | C2 | C23×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups; only the 477 normal subgroups are shown.
This group has 1019 conjugacy classes of subgroups — too many to draw. Only its 477 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)(3,14)(4,10)(5,11)(6,24)(7,28)(8,13)(9,17)(12,26)(15,39)(16,40)(18,41)(19,34)(20,35)(21,48)(22,51)(23,38)(25,31)(27,53)(29,37)(30,44)(32,54)(33,45)(36,50)(42,60)(43,61)(46,63)(47,58)(49,56)(52,59)(55,62)(57,64);(1,3)(2,9)(4,15)(5,16)(6,25)(7,29)(8,18)(10,30)(11,31)(12,37)(13,33)(14,24)(17,26)(19,42)(20,43)(21,49)(22,52)(23,45)(27,54)(28,32)(34,55)(35,56)(36,59)(38,40)(39,48)(41,53)(44,50)(46,64)(47,62)(51,57)(58,61)(60,63);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22)(8,19)(9,30)(11,35)(12,36)(13,34)(14,39)(16,43)(17,44)(18,42)(23,47)(24,48)(25,49)(26,50)(27,46)(28,51)(29,52)(31,56)(32,57)(33,55)(37,59)(38,58)(40,61)(41,60)(45,62)(53,63)(54,64);(4,19)(10,34)(15,42)(20,22)(21,46)(30,55)(35,51)(36,58)(39,60)(43,52)(44,62)(47,50)(48,63)(49,64)(56,57)(59,61);(1,5,8,7)(2,11,13,28)(3,16,18,29)(4,20,19,22)(6,23,27,26)(9,31,33,32)(10,35,34,51)(12,24,38,53)(14,40,41,37)(15,43,42,52)(17,25,45,54)(21,47,46,50)(30,56,55,57)(36,48,58,63)(39,61,60,59)(44,49,62,64)Polycyclic presentation:
2^8|e^2=h,f^2=gh,g^2=h,b^a=bf,f^a=fg,g^a=gh,f^b=fg,g^b=gh,d^c=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #62635⋊C3 |
| maximal subgroup of | #152167, #152168, #152173, … |
| maximal normal subgroup of | #152167, #152168, #152173, … |
| maximal quotient of | #7622496, #7622536, #7622605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55623);