Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
774 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)(5,6,12,19)(7,20)(8,27)(9,15,21,18)(10,17,30,36)(11,16,28,35)(13,25)(14,23)(22,32)(24,31)(26,33)(29,34);(5,7,10,23,9,22,11,29,12,26,30,25,21,8,28,24)(6,13,16,20,18,34,17,27,19,14,35,33,15,31,36,32);(2,4)(5,8)(6,14)(7,21)(9,26)(10,23)(11,24)(12,22)(13,19)(15,31)(16,32)(17,33)(18,34)(20,36)(25,30)(27,35)(28,29);(5,9,12,21)(6,15,19,18)(7,22,26,8)(10,11,30,28)(13,31,14,34)(16,36,35,17)(20,32,33,27)(23,29,25,24)Polycyclic presentation:
2^9|a^2=ef,b^2=e,d^2=i,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249933, #72249941, #72249962, … |
| maximal normal subgroup of | #72249933, #72249941, #72249962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406502);