Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1812 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8608 |
| prime radical | #525 | D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,13,14,36)(4,18)(5,22,23,9)(6,25)(7,12)(8,15,29,38)(10,20)(11,26)(16,34)(17,44,40,37)(19,47,48,45)(21,46)(24,32,53,52)(27,55)(28,41)(30,58,54,42)(31,49,51,63)(33,50)(35,39)(43,60)(56,62)(57,64)(59,61);(1,3,4,15,6,17,20,42,7,14,21,38,26,40,50,58)(2,8,10,13,11,30,18,37,12,29,33,36,25,54,46,44)(5,16,19,41,24,43,49,62,23,39,48,55,53,61,63,64)(9,28,31,35,32,57,45,60,22,27,51,34,52,56,47,59);(2,9)(3,14)(8,27)(10,31)(11,32)(12,22)(13,34)(15,38)(16,39)(17,40)(18,45)(25,52)(28,29)(30,56)(33,51)(35,36)(37,59)(41,55)(42,58)(43,61)(44,60)(46,47)(54,57)(62,64)Polycyclic presentation:
2^9|a^2=f,b^2=d,d^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,f^b=fi,f^c=fh,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287362, #72343002 |
| maximal normal subgroup of | #72287362, #72343002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41944);