Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,17,26,15)(7,18,21,12,20,29,35,28)(8,13,23,27,22,30,36,31)(9,14,24,33,25,34,19,32);(5,7,10,21,11,20,26,35)(6,12,15,18,17,28,16,29)(8,19,23,9,22,24,36,25)(13,14,31,32,30,34,27,33);(2,4)(5,8)(6,13)(7,19)(9,21)(10,23)(11,22)(12,14)(15,31)(16,27)(17,30)(18,32)(20,24)(25,35)(26,36)(28,34)(29,33);(5,9,10,24,11,25,26,19)(6,14,16,33,17,34,15,32)(7,8,21,23,20,22,35,36)(12,27,29,30,28,31,18,13);(7,20)(8,22)(12,28)(13,30)(18,29)(21,35)(23,36)(27,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=di,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72265535, #72307041, #72307328, … |
| maximal normal subgroup of | #72265535, #72307041, #72307328, … |
| maximal quotient of | #72310134, #72310152, #72343949 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280097);