Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,23)(16,29)(17,32)(18,21)(19,27)(20,30)(22,24)(25,39)(26,35)(28,34)(31,36)(33,40)(37,41)(38,44)(42,43);(6,9)(7,11)(13,15,24,26,18,28,17,27)(14,19,32,34,21,35,22,23)(16,25,38,40,30,41,31,42)(20,33,44,39,29,43,36,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,25)(17,31)(18,30)(19,33)(21,29)(22,36)(23,37)(24,38)(26,40)(27,42)(28,41)(32,44)(34,39)(35,43);(6,9)(10,12)(15,26)(17,24)(19,34)(22,32)(23,35)(25,40)(27,28)(31,38)(33,39)(36,44)(37,43)(41,42);(5,7,8,11)(6,10,9,12)(14,21)(16,30)(19,35)(22,32)(23,34)(25,41)(31,38)(40,42)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7056563);