Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232766 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2397 |
| prime radical | #10012 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
717 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 717 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2,3,7,4,6,8,5)(9,10,13,17,14,18,24,31)(11,19,21,28,22,33,27,15)(12,23,25,39,26,16,30,32)(20,35,36,29,37,34,40,38);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,27)(14,22)(16,29)(17,33)(18,28)(19,31)(21,24)(23,38)(25,40)(26,37)(30,36)(32,35)(34,39);(2,5)(3,8)(6,7)(15,28)(19,33)(29,38)(34,35);(2,6)(5,7)(9,12,24,30,14,26,13,25)(10,16,31,39,18,23,17,32)(11,20,21,36,22,37,27,40)(15,29,19,34,28,38,33,35)Polycyclic presentation:
2^9|a^2=e,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283833, C2×#244127, #72307194, … |
| maximal normal subgroup of | #72283833, C2×#244127, #72307194, … |
| maximal quotient of | C2×#244127 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151323);