Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23848 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10,12,14)(11,15,16,13);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,18)(19,26)(20,23)(21,25)(22,27)(24,29)(28,31)(30,32);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,22)(20,28)(21,27)(23,30)(24,31)(25,26)(29,32);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,23)(20,29)(22,30)(24,25)(26,31)(28,32);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,27)(22,26)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fg,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72251240, #72251258, #72256957, … |
| maximal normal subgroup of | #72251240, #72251258, #72256957, … |
| maximal quotient of | #72308317, #72308793 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7497152);