Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4607 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)(5,6,9,16,11,17,10,15)(7,18,23,28,21,29,22,12)(8,14,25,13,26,30,24,31)(19,34,36,32,35,27,20,33);(5,7)(6,12)(8,19)(9,23)(10,22)(11,21)(13,27)(14,32)(15,29)(16,18)(17,28)(20,24)(25,36)(26,35)(30,33)(31,34);(2,4)(5,8)(6,13)(7,19)(9,24)(10,25)(11,26)(12,27)(14,16)(15,30)(17,31)(18,32)(20,23)(21,35)(22,36)(28,34)(29,33);(5,8)(6,14)(7,20)(9,25)(10,24)(11,26)(12,27)(13,16)(15,31)(17,30)(18,33)(19,23)(21,36)(22,35)(28,34)(29,32);(7,21)(12,28)(18,29)(19,35)(20,36)(22,23)(27,34)(32,33)Polycyclic presentation:
2^9|a^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,e^b=ei,f^b=fi,g^c=gi,h^c=hi,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279230, #72312189, #72312221, … |
| maximal normal subgroup of | #72279230, #72312189, #72312221, … |
| maximal quotient of | #102×AGL(1,ℤ/8), #72309571, #72322064 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286407);