Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130780 |
| 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 | #61081 | 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 | |
791 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 791 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:
(1,2,7,14,6,31,33,49,9,15,35,50,8,13,23,47)(3,16,20,53,36,52,55,63,21,54,60,61,19,45,34,38)(4,22,25,46,32,43,30,12,26,44,29,62,24,11,5,27)(10,37,40,58,39,17,48,28,41,56,51,18,42,57,64,59);(1,3)(2,10)(4,17)(5,28)(6,19)(7,34)(8,36)(9,21)(11,38)(12,45)(13,39)(14,48)(15,41)(16,46)(18,29)(20,23)(22,61)(24,56)(25,58)(26,57)(27,52)(30,59)(31,42)(32,37)(33,60)(35,55)(40,49)(43,63)(44,53)(47,51)(50,64)(54,62);(1,4,9,26)(2,11,15,43)(3,17,21,57)(5,23,30,33)(6,32,8,24)(7,29,35,25)(10,38,41,63)(12,14,27,50)(13,44,31,22)(16,51,54,40)(18,55,58,34)(19,37,36,56)(20,59,60,28)(39,53,42,61)(45,48,52,64)(46,47,62,49)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,e^2=i,f^2=hi,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,e^c=ei,f^c=fgi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72307133, #72307138, #72307140, … |
| maximal normal subgroup of | #72307133, #72307138, #72307140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37979);