Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45134 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2189 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(7,11)(10,12)(13,14,20,23,19,24,18,17)(15,25,37,21,29,35,28,27)(16,22,32,31,34,40,33,36)(26,41,42,38,30,44,43,39);(6,9)(7,11)(13,15,19,29)(14,21,24,27)(16,30,34,26)(17,35,23,25)(18,37,20,28)(22,39,40,38)(31,44,36,41)(32,42,33,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,34)(14,22,24,40)(15,26,29,30)(17,36,23,31)(18,33,20,32)(21,38,27,39)(25,41,35,44)(28,43,37,42);(6,9)(10,12)(13,17,18,24,19,23,20,14)(15,27,28,35,29,21,37,25)(16,31,33,22,34,36,32,40)(26,38,43,41,30,39,42,44);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,23,24,17)(15,28,29,37)(16,32,34,33)(21,35,27,25)(22,36,40,31)(26,42,30,43)(38,41,39,44)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053319);