Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×#509 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,17)(11,15,20,28)(12,16,25,32)(13,26,24,19)(18,34,33,23)(21,31,35,27)(22,39,36,30)(29,37,40,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,22)(14,20)(16,31)(17,28)(18,29)(19,30)(23,38)(24,36)(25,35)(26,39)(27,32)(33,40)(34,37);(10,16)(11,20)(12,23)(13,24)(15,27)(17,32)(18,19)(21,37)(25,34)(26,33)(28,31)(29,39)(30,40)(35,38);(2,5)(6,8)(10,17)(15,28)(16,32)(18,33)(19,26)(27,31)(29,40)(30,39);(1,3,4,7)(2,6,5,8)(10,17)(15,28)(16,32)(18,33)(19,26)(27,31)(29,40)(30,39)Polycyclic presentation:
2^9|a^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286569, #72286644, #72286648, … |
| maximal normal subgroup of | #72286569, #72286644, #72286648, … |
| maximal quotient of | #72343301, #72343370, #72343408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531207);