Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #326502 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2563 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
500 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(1,2,5,7,3,6,4,8)(9,10,14,19)(11,20)(12,25,26,30)(13,15,28,18)(16,24)(17,33,34,27)(21,29)(22,37)(23,32)(31,36)(35,38)(39,40);(2,6)(7,8)(10,15,19,18)(11,21,24,23)(12,26)(17,25,34,30)(22,38,39,36)(27,33)(31,35)(37,40);(2,7)(4,5)(6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,29)(17,31)(18,32)(19,20)(21,28)(25,40)(26,39)(27,38)(30,37)(33,36)(34,35);(2,6)(7,8)(9,12,13,27,14,26,28,33)(10,17,18,25,19,34,15,30)(11,22,23,38,24,39,21,36)(16,31,32,40,20,35,29,37)Polycyclic presentation:
2^9|a^2=efg,b^2=eghi,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,f^c=fi,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 233706);