Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2397 |
| prime radical | #10012 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
723 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 723 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,3,7,5,8,4,6)(9,10,13,17,14,18,26,32)(11,19,21,27,22,31,25,15)(12,16,23,29,24,30,37,40)(20,33,34,38,35,39,36,28);(2,6)(3,4)(7,8)(9,11)(10,15)(12,20)(13,25)(14,22)(16,28)(17,31)(18,27)(19,32)(21,26)(23,36)(24,35)(29,39)(30,38)(33,40)(34,37);(2,7)(3,4)(6,8)(15,27)(19,31)(28,38)(33,39);(9,12,13,23,14,24,26,37)(10,16,17,29,18,30,32,40)(11,20,25,36,22,35,21,34)(15,28,31,39,27,38,19,33)Polycyclic presentation:
2^9|a^2=e,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72283833, C2×#244113, #72307191, … |
| maximal normal subgroup of | #72283833, C2×#244113, #72307191, … |
| maximal quotient of | C2×#244113 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151309);