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 | #60806 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
872 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)(3,11,14,26)(4,15,16,31)(5,18,20,9)(6,10)(7,13,22,29)(8,17)(12,24,19,27)(21,28,23,30)(25,32);(1,3)(2,7)(4,13)(5,19)(6,12)(8,24)(9,22)(10,21)(11,25)(14,20)(15,27)(16,28)(17,29)(18,23)(26,31)(30,32);(4,16)(7,21)(8,25)(13,28)(15,31)(17,32)(22,23)(29,30);(3,12)(7,21)(11,24)(13,28)(14,19)(22,23)(26,27)(29,30)Polycyclic presentation:
2^9|a^2=f,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 | #72213429, #72216447, #72219528, … |
| maximal normal subgroup of | #72213429, #72216447, #72219528, … |
| maximal quotient of | C2×#512647, #72284600, #72284654, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419843);