Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,12)(8,13)(9,14,22,11)(10,15)(16,26)(17,27,23,24)(18,28)(19,29)(20,30,34,25)(21,31)(32,36,33,35);(6,11)(9,22)(12,24)(13,25)(14,15)(17,23)(20,34)(26,35)(27,28)(29,36)(30,31)(32,33);(2,4)(5,7,10,18)(6,12,15,28)(8,19,21,16)(9,23,22,17)(11,24,14,27)(13,29,31,26)(20,32,34,33)(25,36,30,35);(5,8,10,21)(6,13,15,31)(7,16,18,19)(9,20,22,34)(11,25,14,30)(12,26,28,29)(17,32,23,33)(24,35,27,36)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fhi,g^b=ghi,d^c=di,e^c=eh,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306598, #72306748, #72306753, … |
| maximal normal subgroup of | #72306598, #72306748, #72306753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400827);