Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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 |
| prime radical | C2×#3267 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,28)(21,34)(26,35)(27,37)(29,36)(31,38)(32,40)(33,39)(41,42)(43,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,28,19,29,18,27)(14,20,23,25,24,37,22,36)(16,31,33,42,32,26,34,43)(21,38,40,44,39,35,30,41);(7,12)(9,10)(13,16,18,34,19,32,17,33)(14,21,22,30,24,39,23,40)(15,26,27,42,29,31,28,43)(20,35,36,44,37,38,25,41);(5,7)(6,10)(8,12)(9,11)(16,32)(21,39)(26,31)(30,40)(33,34)(35,38)(41,44)(42,43);(7,12)(9,10)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,33,32,34)(20,25,37,36)(21,30,39,40)(26,42,31,43)(35,41,38,44)Polycyclic presentation:
2^9|b^2=fhi,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6514382);