Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33638 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,11,14,25)(7,16)(8,19)(9,22)(12,26,27,33)(13,20,15,24)(17,30)(18,32)(21,34)(23,29,35,28);(5,6,9,13,10,14,22,15)(7,12,16,28,17,27,30,29)(8,20,19,11,21,24,34,25)(18,33,31,23,32,26,36,35);(1,2,3,4)(5,7,9,16,10,17,22,30)(6,12,13,28,14,27,15,29)(8,18,19,31,21,32,34,36)(11,23,24,26,25,35,20,33);(7,17)(12,27)(16,30)(18,32)(23,35)(26,33)(28,29)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=eh,g^b=gi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226793, #72226795, #72226796, … |
| maximal normal subgroup of | #72226793, #72226795, #72226796, … |
| maximal quotient of | #72285455, #72310109, #72319644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402274);