Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1362 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)(5,6,11,17,9,15,26,16,13,18,27,28,12,7,10,20)(8,21,24,19,30,35,23,33,25,34,36,31,22,32,29,14);(5,7,9,18,13,15,12,6)(8,19,22,33,25,31,30,14)(10,20,11,28,26,16,27,17)(21,29,35,36,34,23,32,24);(2,4)(5,8)(6,14)(7,19)(9,22)(10,24)(11,29)(12,30)(13,25)(15,31)(16,34)(17,32)(18,33)(20,21)(23,27)(26,36)(28,35);(5,9,13,12)(6,15,18,7)(8,22,25,30)(10,11,26,27)(14,31,33,19)(16,28,20,17)(21,32,34,35)(23,24,29,36)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,d^2=i,e^2=h,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249862, #72249864, #72249902, … |
| maximal normal subgroup of | #72249862, #72249864, #72249902, … |
| maximal quotient of | #72265643, #72280313, #72285367 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401741);