Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#3287 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
955 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 955 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10)(11,20)(12,17)(13,25)(14,29)(15,18)(16,22)(19,27)(21,36)(23,37)(24,34)(26,31)(28,39)(30,35)(32,38)(33,40);(2,6)(3,4)(7,8)(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,31)(18,32)(19,35)(20,25)(22,27)(24,38)(28,40)(29,33)(34,36)(37,39);(2,7)(3,4)(6,8)(9,12,15,14)(10,17,18,29)(11,21,23,30)(13,24,28,27)(16,31,32,33)(19,25,34,39)(20,36,37,35)(22,26,38,40);(10,18)(16,32)(17,29)(19,34)(20,37)(25,39)(31,33)(35,36)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309988, #72309991, #72309993, … |
| maximal normal subgroup of | #72309988, #72309991, #72309993, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396143);