Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144119 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2011 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33717 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
533 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 533 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 3 generators on 40 points:
(1,2)(3,6)(4,7)(5,8)(9,10,16,22)(11,23,28,40)(12,29,32,21)(13,18,15,20)(14,33,31,19)(17,24,38,25)(26,37,34,35)(27,36,30,39);(2,6)(3,4)(7,8)(9,11)(10,17)(12,30)(13,26)(14,24)(15,34)(16,28)(18,39)(19,35)(20,36)(21,40)(22,38)(23,29)(25,31)(27,32)(33,37);(2,7)(3,4)(6,8)(10,18)(11,24)(12,31)(13,15)(14,32)(17,35)(19,33)(20,22)(23,36)(25,28)(26,30)(27,34)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=h,f^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fgh,g^b=gi,d^c=dg,e^c=eh,f^c=fgh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224919, #72224925, #72224926 |
| maximal normal subgroup of | #72224919, #72224925, #72224926 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51316);