Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501868 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2190 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1753 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,20)(9,12)(10,25)(11,17)(13,24)(14,16)(18,30)(19,34)(21,28)(22,33)(23,32)(26,31)(27,36)(29,35);(1,2)(3,4)(5,7,16,18,9,19,10,26)(6,11,25,28,12,27,14,32)(8,21,35,17,22,23,24,36)(13,30,33,15,29,31,20,34);(2,4)(6,12)(8,22)(11,27)(14,25)(17,36)(21,23)(24,35)(28,32);(5,7,9,19)(6,11,12,27)(8,23,22,21)(10,26,16,18)(13,31,29,30)(14,32,25,28)(15,33,34,20)(17,35,36,24)Polycyclic presentation:
2^9|b^2=fh,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=ehi,f^b=fi,h^b=hi,e^c=ei,e^d=eh,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72250011, #72252775, #72252788, … |
| maximal normal subgroup of | #72250011, #72252775, #72252788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409065);