Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C2×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,8,6)(9,10)(11,19,21,36)(12,23)(13,27)(14,16)(15,20,30,39)(17,25)(18,29)(22,33,28,37)(24,34,40,31)(26,35)(32,38);(1,3)(2,5)(4,6)(7,8)(9,11)(10,15)(12,24)(13,28)(14,21)(16,30)(17,33)(18,34)(19,35)(20,38)(22,29)(23,36)(25,39)(26,40)(27,31)(32,37);(2,3)(6,7);(10,16)(15,30)(17,32)(18,27)(19,36)(23,35)(31,34)(33,37)Polycyclic presentation:
2^9|a^2=f,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ch,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72287736, #72287755, #72287768, … |
| maximal normal subgroup of | #72287736, #72287755, #72287768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388540);