Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42800 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(7,11)(10,12)(13,14)(15,24)(16,22)(17,18)(19,36)(20,23)(21,28)(25,41)(26,27)(29,34)(30,44)(31,32)(33,40)(35,37)(38,42)(39,43);(6,9)(7,11)(13,15,20,29)(14,21,23,26)(16,30,33,25)(17,34,36,24)(18,28,19,27)(22,39,40,38)(31,44,35,41)(32,42,37,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,33)(14,22,23,40)(15,25,29,30)(17,35,36,31)(18,37,19,32)(21,38,26,39)(24,41,34,44)(27,43,28,42);(6,9)(10,12)(13,17,20,36)(14,18,23,19)(15,26,29,21)(16,31,33,35)(22,32,40,37)(24,27,34,28)(25,38,30,39)(41,42,44,43);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,17,23,36)(15,27,29,28)(16,32,33,37)(21,24,26,34)(22,31,40,35)(25,42,30,43)(38,44,39,41)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050357);