Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2211 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)(15,24)(16,22)(17,18)(19,32)(20,23)(21,27)(25,38)(26,35)(28,39)(29,33)(30,36)(31,37)(34,40)(41,43)(42,44);(6,9)(7,11)(13,15,19,27,20,28,18,26)(14,21,17,24,23,35,32,39)(16,25,30,40,31,41,33,42)(22,34,29,38,37,44,36,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,25)(17,29)(18,33)(19,30)(20,31)(21,34)(23,37)(24,38)(26,42)(27,40)(28,41)(32,36)(35,44)(39,43);(6,9)(10,12)(13,17,20,32)(14,18,23,19)(15,21,28,35)(16,29,31,36)(22,33,37,30)(24,27,39,26)(25,34,41,44)(38,40,43,42);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,17,23,32)(15,26,28,27)(16,30,31,33)(21,24,35,39)(22,36,37,29)(25,40,41,42)(34,43,44,38)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7050902);