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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,24)(15,25,27,37)(16,30,32,40)(17,23,18,22)(20,29,36,28)(21,33,39,34)(26,35,31,38)(41,42,44,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,29,19,27,17,28)(14,20,23,37,24,36,22,25)(16,31,34,42,32,26,33,43)(21,38,30,41,39,35,40,44);(7,12)(9,10)(13,16,17,33,19,32,18,34)(14,21,22,40,24,39,23,30)(15,26,28,42,27,31,29,43)(20,35,25,41,36,38,37,44);(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)(15,27)(20,36)(25,37)(26,31)(28,29)(35,38)(41,44)(42,43)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=gi,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, 6514377);