Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C2×#578 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10)(11,20)(12,18)(13,19)(14,27)(15,17)(16,22)(21,35)(23,31)(24,33)(25,32)(26,34)(28,29)(30,37)(36,39)(38,40);(9,11,15,23)(10,16,17,29)(12,21,25,38)(13,22,14,28)(18,30,32,39)(19,31,27,20)(24,36,26,37)(33,35,34,40);(1,3,4,8,5,2,6,7)(10,17)(16,29)(18,32)(19,27)(20,31)(30,39)(33,34)(35,40);(2,3)(7,8)(9,12)(10,18)(11,21)(13,26)(14,24)(15,25)(16,30)(17,32)(19,34)(20,35)(22,37)(23,38)(27,33)(28,36)(29,39)(31,40);(9,13,15,14)(10,19,17,27)(11,22,23,28)(12,24,25,26)(16,31,29,20)(18,33,32,34)(21,36,38,37)(30,35,39,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=di,f^a=fh,f^b=fh,d^c=di,e^d=eh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72322843, #72322853, #72322898, … |
| maximal normal subgroup of | #72322843, #72322853, #72322898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252292);