Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| 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 | #33179 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2090 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)(7,12)(8,10)(9,11)(13,14)(15,27)(16,33)(17,23)(18,24)(19,25)(20,26)(21,30)(22,29)(28,38)(31,40)(32,39)(34,42)(35,41)(36,43)(37,44);(1,2,3,4)(7,8)(10,12)(13,15,18,30,20,32,19,31)(14,21,25,27,26,40,24,39)(16,28,34,36,35,17,29,37)(22,38,41,44,42,23,33,43);(5,7,9,8)(6,10,11,12)(13,16,19,29,20,35,18,34)(14,22,24,33,26,42,25,41)(15,28,31,37,32,17,30,36)(21,38,39,43,40,23,27,44);(5,8,9,7)(6,10,11,12)(13,17,20,28)(14,23,26,38)(15,29,32,34)(16,31,35,30)(18,37,19,36)(21,33,40,41)(22,39,42,27)(24,44,25,43);(6,11)(10,12)(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,29,35,34)(17,36,28,37)(21,39,40,27)(22,41,42,33)(23,43,38,44)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390688);