Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8375 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #61064 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
798 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 798 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(1,2,5,11,6,12,27,40,7,9,26,33,21,34,53,59)(3,13,16,42,17,35,24,55,15,43,46,60,28,48,56,64)(4,18,22,39,23,49,54,63,20,10,25,36,52,37,57,62)(8,29,31,14,32,45,61,19,30,51,58,44,41,47,38,50);(1,3)(2,8)(4,19)(5,24)(6,28)(7,15)(9,30)(10,35)(11,38)(12,41)(13,37)(14,23)(16,27)(17,21)(18,48)(20,50)(22,47)(25,45)(26,56)(29,57)(31,59)(32,34)(33,61)(36,42)(39,60)(40,58)(43,49)(44,52)(46,53)(51,54)(55,63)(62,64);(2,9)(4,20)(8,30)(12,34)(14,44)(19,50)(22,25)(23,52)(29,51)(32,41)(36,39)(42,60)(45,47)(54,57)(55,64)(62,63)Polycyclic presentation:
2^9|a^2=f,d^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,f^c=fg,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72307706, #72307714, #72307736, … |
| maximal normal subgroup of | #72307706, #72307714, #72307736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37980);