Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130776 |
| 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 | #61054 | 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 | |
805 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 805 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:
(1,2,6,13,7,10,30,38,8,12,29,41,4,19,21,43)(3,14,17,47,15,39,27,57,18,46,51,61,31,45,58,64)(5,22,24,44,25,52,54,63,26,11,28,40,20,37,53,62)(9,32,35,16,36,48,60,23,34,55,59,49,33,50,42,56);(1,3)(2,9)(4,15)(5,23)(6,27)(7,31)(8,18)(10,33)(11,39)(12,34)(13,42)(14,37)(16,25)(17,30)(19,36)(20,49)(21,51)(22,45)(24,50)(26,56)(28,48)(29,58)(32,53)(35,43)(38,59)(40,47)(41,60)(44,61)(46,52)(54,55)(57,63)(62,64);(1,4,8,7)(2,10,12,19)(3,15,18,31)(5,20,26,25)(6,21,29,30)(9,33,34,36)(11,37,22,52)(13,43,41,38)(14,45,46,39)(16,23,49,56)(17,27,51,58)(24,53,28,54)(32,48,55,50)(35,60,59,42)(40,63,44,62)(47,57,61,64)Polycyclic presentation:
2^9|a^2=f,c^2=i,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,f^b=fgh,h^b=hi,f^c=fg,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72307076, #72307079, #72307095, … |
| maximal normal subgroup of | #72307076, #72307079, #72307095, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37973);