Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | #42679 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2004 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)(6,9)(7,11)(13,14)(15,20)(16,21)(17,19)(18,22)(23,36)(24,33)(25,31)(26,35)(27,41)(28,38)(29,32)(30,37)(34,39)(40,44)(42,43);(5,6)(7,10)(8,9)(11,12)(14,19)(15,23)(16,27)(18,24)(20,34)(21,37)(25,39)(29,30)(31,36)(32,41)(35,38)(40,42);(1,2,3,4)(6,9)(10,12)(13,15,24,39,17,25,18,23)(14,20,33,34,19,31,22,36)(16,28,27,40,29,26,30,42)(21,38,41,44,32,35,37,43);(5,7,8,11)(6,10,9,12)(14,19)(15,25)(21,32)(22,33)(23,39)(26,28)(37,41)(40,42);(13,16,24,27,17,29,18,30)(14,21,33,41,19,32,22,37)(15,26,39,42,25,28,23,40)(20,35,34,43,31,38,36,44)Polycyclic presentation:
2^9|c^2=fgh,d^2=fi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724296);