Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×#509 |
| 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 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2043 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)(5,6,10,15)(7,16,18,27)(8,21,24,30)(9,12,23,14)(11,26,28,19)(13,31,33,22)(17,34,35,29)(20,36,25,32);(5,7)(6,11)(8,17)(9,25)(10,18)(12,29)(13,32)(14,34)(15,28)(16,21)(19,22)(20,23)(24,35)(26,31)(27,30)(33,36);(2,4);(6,12)(7,17)(8,22)(9,23)(11,27)(14,15)(16,28)(18,35)(19,20)(21,30)(24,31)(25,26)(29,32)(34,36);(7,18)(11,28)(16,27)(17,35)(19,26)(20,25)(29,34)(32,36)Polycyclic presentation:
2^9|a^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,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 | #72250863, #72250867, #72250895, … |
| maximal normal subgroup of | #72250863, #72250867, #72250895, … |
| maximal quotient of | #72287679, #72309422, #72309684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530817);