Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1084 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)(7,18)(8,13)(9,15)(10,25)(11,27)(12,17)(14,21)(16,23)(19,32)(20,33)(22,30)(24,28)(26,36)(29,35)(31,34);(6,13)(7,19)(8,9)(10,24)(11,23)(14,29)(15,16)(17,25)(18,31)(20,26)(21,34)(22,35)(27,28)(30,33);(2,4)(5,7,8,19,23,34,24,35,12,20,10,26,11,22,9,21)(6,14,15,30,27,36,25,33,17,29,28,31,16,32,13,18);(7,20)(14,29)(18,33)(19,26)(21,35)(22,34)(30,31)(32,36)Polycyclic presentation:
2^9|c^2=e,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,d^c=di,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250011, #72250198, #72250204, … |
| maximal normal subgroup of | #72250011, #72250198, #72250204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406379);