Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512647 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #124 | C24 |
| prime radical | #60808 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C23×C4 ◁ #3449 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11)(3,9,15,23)(4,8,17,21)(6,10,18,25)(7,12,19,26)(13,20,27,30)(14,24,28,32)(16,22,29,31);(2,8)(4,6)(5,18)(9,22)(10,12)(13,16)(15,29)(17,19)(20,24)(21,25)(23,32)(27,28);(1,3)(2,9)(4,13)(5,15)(6,16)(7,14)(8,20)(10,22)(11,23)(12,24)(17,27)(18,29)(19,28)(21,30)(25,31)(26,32);(1,4,7,6)(2,8,12,10)(3,13,14,16)(5,17,19,18)(9,20,24,22)(11,21,26,25)(15,27,28,29)(23,30,32,31)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=h,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=eh,f^b=fghi,g^b=gi,e^c=ei,f^e=fhi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72284730, #72287234, #72292692, … |
| maximal normal subgroup of | #72284730, #72287234, #72292692, … |
| maximal quotient of | C2×#512649, #72292960, #72292961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419845);