Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2238 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)(19,34)(20,23)(21,27)(25,41)(26,35)(28,38)(29,44)(30,31)(32,40)(33,36)(37,43)(39,42);(6,9)(7,11)(13,15,18,35,20,28,19,27)(14,21,34,38,23,26,17,24)(16,29,36,43,32,25,31,42)(22,39,30,41,40,37,33,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,32)(14,22,23,40)(15,25,28,29)(17,33,34,30)(18,36,19,31)(21,37,26,39)(24,41,38,44)(27,43,35,42);(6,9)(10,12)(13,17,20,34)(14,18,23,19)(15,26,28,21)(16,30,32,33)(22,31,40,36)(24,35,38,27)(25,37,29,39)(41,43,44,42);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,17,23,34)(15,27,28,35)(16,31,32,36)(21,38,26,24)(22,30,40,33)(25,42,29,43)(37,41,39,44)Polycyclic presentation:
2^9|b^2=fi,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=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7051009);