Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130786 |
| 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 | #61091 | 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 | |
789 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 789 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:
(1,2,7,14,6,13,30,49,8,16,35,45,9,15,25,51)(3,17,21,54,20,53,34,37,22,55,62,63,23,44,57,64)(4,24,27,46,33,11,31,42,28,48,32,47,26,43,5,12)(10,36,39,19,38,18,56,58,41,61,52,60,40,59,50,29);(1,3,8,22)(2,10,16,41)(4,18,28,59)(5,29,31,58)(6,23,9,20)(7,34,35,57)(11,37,43,64)(12,44,42,53)(13,40,15,38)(14,50,45,56)(17,46,55,47)(19,27,60,32)(21,25,62,30)(24,63,48,54)(26,61,33,36)(39,49,52,51);(1,4,8,28)(2,11,16,43)(3,18,22,59)(5,30,31,25)(6,33,9,26)(7,27,35,32)(10,37,41,64)(12,45,42,14)(13,48,15,24)(17,52,55,39)(19,34,60,57)(20,61,23,36)(21,58,62,29)(38,63,40,54)(44,56,53,50)(46,51,47,49)Polycyclic presentation:
2^9|a^2=f,b^2=i,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,e^b=ei,f^b=fgh,h^b=hi,d^c=di,e^c=ei,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72307076, #72307103, #72307112, … |
| maximal normal subgroup of | #72307076, #72307103, #72307112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37989);