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×#575 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1753 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,12)(7,15,19,36)(8,20,22,35)(10,25,16,14)(11,27,28,17)(13,31,30,24)(18,32,26,33)(21,29,23,34);(1,2)(3,4)(5,7,16,18,9,19,10,26)(6,11,25,29,12,28,14,34)(8,21,31,17,22,23,24,27)(13,32,35,15,30,33,20,36);(2,4)(6,12)(8,22)(11,28)(14,25)(17,27)(21,23)(24,31)(29,34);(5,7,9,19)(6,11,12,28)(8,23,22,21)(10,26,16,18)(13,33,30,32)(14,34,25,29)(15,35,36,20)(17,31,27,24)Polycyclic presentation:
2^9|a^2=i,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 | #72321029 |
| maximal normal subgroup of | #72321029 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409067);