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 | #8609 |
| prime radical | #532 | D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
291 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 291 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:
(1,2,6,13,8,14,7,29)(3,15,20,42,17,43,21,51)(4,22,24,39,25,54,16,11)(5,12,27,37,26,10,28,36)(9,30,32,46,33,44,41,18)(19,40,49,62,47,61,50,63)(23,53,45,38,55,52,56,35)(31,58,59,64,34,57,60,48);(1,3,16,44,7,21,25,46,8,17,24,30,6,20,4,18)(2,9,22,42,13,32,39,43,14,33,54,51,29,41,11,15)(5,19,45,58,27,49,55,64,26,47,56,57,28,50,23,48)(10,34,52,62,37,59,38,40,12,31,53,63,36,60,35,61);(2,10)(3,17)(5,26)(9,31)(11,35)(12,14)(13,37)(15,40)(18,46)(20,21)(22,52)(23,55)(27,28)(29,36)(30,44)(32,60)(33,34)(38,39)(41,59)(42,63)(43,61)(45,56)(51,62)(53,54)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,e^c=ei,f^c=fhi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287240, #72287291, #72321395, … |
| maximal normal subgroup of | #72287240, #72287291, #72321395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41974);