Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,21)(9,12)(10,13)(11,19)(14,22)(15,23)(17,32)(18,26)(20,35)(24,29)(25,31)(27,33)(28,34)(30,36);(5,7)(6,11)(8,17)(9,25)(10,18)(12,27)(13,28)(14,31)(15,32)(16,21)(19,22)(20,23)(24,33)(26,29)(30,35)(34,36);(2,4);(6,12)(7,17)(8,22)(9,23)(11,26)(13,15)(16,28)(18,33)(19,20)(21,29)(24,36)(25,34)(27,31)(32,35);(6,13)(7,18)(12,15)(14,30)(17,33)(19,34)(20,25)(21,29)Polycyclic presentation:
2^9|f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250895, #72315814, #72315824, … |
| maximal normal subgroup of | #72250895, #72315814, #72315824, … |
| maximal quotient of | #72319537, #72319542, #72319556, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530822);