Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33469 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1210 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)(7,18,22,32)(8,14)(9,28)(10,29)(11,25)(12,17)(13,21,31,23)(15,27)(16,26)(19,33,30,36)(20,34,24,35);(2,4)(5,7,10,23,11,24,26,36,12,22,9,21,8,20,27,33)(6,13,16,32,14,30,29,35,17,31,15,18,25,19,28,34);(5,6)(7,19,22,30)(8,25)(9,16)(10,15)(11,14)(12,17)(13,20,31,24)(18,33,32,36)(21,34,23,35)(26,28)(27,29);(5,8,12,11)(6,14,17,25)(7,20,22,24)(9,26,10,27)(13,30,31,19)(15,28,16,29)(18,34,32,35)(21,36,23,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=h,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72226791, #72226795, #72226798, … |
| maximal normal subgroup of | #72226791, #72226795, #72226798, … |
| maximal quotient of | #72309756, #72309785, #72309840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400275);