Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501164 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,12)(9,21,23,13)(10,14)(11,17,25,34)(16,29)(18,30)(19,28)(20,26,22,27)(24,32,36,33)(31,35);(1,2)(3,4)(5,7,10,18)(6,11,14,25)(8,16,19,31)(9,17,23,34)(12,24,28,36)(13,15,21,30)(20,32,22,33)(26,35,27,29);(2,4)(5,8)(6,12)(7,16)(9,22)(10,19)(11,24)(13,27)(14,28)(15,29)(17,33)(18,31)(20,23)(21,26)(25,36)(30,35)(32,34);(6,13)(8,19)(11,15)(12,26)(14,21)(16,31)(20,22)(24,35)(25,30)(27,28)(29,36)(32,33)Polycyclic presentation:
2^9|a^2=f,b^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^b=fh,d^c=di,e^c=ehi,f^d=fh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72312319, #72312330, #72312352, … |
| maximal normal subgroup of | #72312319, #72312330, #72312352, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408364);