Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14,19,25)(15,23,28,21)(16,30,33,41)(17,35,29,38)(18,26,20,24)(22,34,40,32)(27,43,44,39)(31,42,36,37);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15)(14,21)(16,31)(17,18)(19,28)(20,29)(22,39)(23,25)(24,35)(26,38)(27,32)(30,37)(33,36)(34,44)(40,43)(41,42);(5,8,9,6)(7,11,12,10)(13,16,20,34,19,33,18,32)(14,22,26,41,25,40,24,30)(15,27,17,36,28,44,29,31)(21,37,35,43,23,42,38,39);(7,12)(10,11)(13,17,18,15,19,29,20,28)(14,23,24,35,25,21,26,38)(16,31,32,44,33,36,34,27)(22,37,30,39,40,42,41,43);(1,3,4,2)(13,18,19,20)(14,24,25,26)(15,17,28,29)(16,32,33,34)(21,35,23,38)(22,30,40,41)(27,36,44,31)(37,43,42,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=h,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511456);