Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623613 |
| 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 | |
2026 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)(17,27)(18,32)(20,36)(24,29)(25,31)(26,34)(28,33)(30,35);(5,7,10,17)(6,11,13,26)(8,18,24,33)(9,25,23,20)(12,28,15,32)(14,31,30,36)(16,29,27,21)(19,35,34,22);(2,4)(7,17)(11,26)(16,27)(18,33)(19,34)(20,25)(28,32)(31,36);(6,12)(7,18)(8,22)(9,23)(11,27)(13,15)(16,26)(17,33)(19,20)(21,29)(24,35)(25,34)(28,31)(32,36);(6,13)(11,26)(12,15)(14,30)(16,27)(21,29)(28,32)(31,36)Polycyclic presentation:
2^9|b^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=df,f^b=fh,h^b=hi,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250895, #72315829, #72315864, … |
| maximal normal subgroup of | #72250895, #72315829, #72315864, … |
| maximal quotient of | #72319574, #72319580, #72319596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530810);