Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152176 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12731 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#3415 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
635 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 635 conjugacy classes of subgroups — too many to draw. Only its 104 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)(3,15,18,40)(4,20)(5,12)(6,28)(7,32)(8,14)(9,16,35,53)(10,17,38,27)(11,24)(13,30)(19,39,55,64)(21,45,54,48)(22,46)(23,33)(25,61)(26,47)(29,49,51,41)(31,42)(34,52,37,59)(36,62,63,57)(43,58)(44,50)(56,60);(1,3)(2,9)(4,21)(5,17)(6,29)(7,27)(8,18)(10,33)(11,41)(12,34)(13,45)(14,35)(15,47)(16,50)(19,30)(20,40)(22,55)(23,57)(24,53)(25,62)(26,54)(28,36)(31,51)(32,37)(38,61)(39,43)(42,63)(44,49)(46,48)(52,60)(56,64)(58,59);(1,3)(2,10)(4,16)(5,27)(6,19)(7,17)(8,18)(9,33)(11,36)(12,40)(13,39)(14,38)(15,32)(20,34)(21,50)(22,51)(23,59)(24,54)(25,52)(26,53)(28,41)(29,30)(31,55)(35,61)(37,47)(42,49)(43,45)(44,63)(46,64)(48,56)(57,58)(60,62)Polycyclic presentation:
2^9|a^2=f,d^2=ghi,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgi,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72224989, #72224997, #72225006, … |
| maximal normal subgroup of | #72224989, #72224997, #72225006, … |
| maximal quotient of | #72227106, C2×#152176 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59372);