Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1052 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)(6,12,14,24)(7,17,19,34)(8,21)(9,23)(10,25)(13,29)(15,26,22,16)(18,20,36,27)(28,35)(30,32);(2,4)(5,6,10,26,11,14,25,16)(7,13,27,32,19,30,20,29)(8,22,21,12,23,15,9,24)(17,33,36,28,34,31,18,35);(5,7)(6,13)(8,17)(9,18)(10,27)(11,19)(12,28)(14,30)(15,31)(16,29)(20,25)(21,36)(22,33)(23,34)(24,35)(26,32);(6,14)(12,24)(13,30)(15,22)(16,26)(28,35)(29,32)(31,33)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^b=di,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ehi,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225127, #72239492, #72249862, … |
| maximal normal subgroup of | #72225127, #72239492, #72249862, … |
| maximal quotient of | #72225090, #72225120, #72286368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401531);