Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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×#3267 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,21)(17,30)(18,23)(19,24)(20,28)(22,32)(26,34)(27,35)(29,38)(31,37)(33,36)(39,40)(41,44)(42,43);(6,9)(7,11)(13,15,18,28)(14,20,23,25)(16,27,19,33)(17,26,32,40)(21,36,24,35)(22,34,30,39)(29,41,31,42)(37,44,38,43);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(17,31)(18,19)(22,38)(23,24)(26,40)(27,33)(29,32)(30,37)(34,39)(35,36);(6,9)(10,12)(13,17,18,32)(14,22,23,30)(15,26,28,40)(16,29,19,31)(20,34,25,39)(21,37,24,38)(27,41,33,42)(35,43,36,44);(14,23)(15,27)(16,19)(20,35)(22,30)(25,36)(26,41)(28,33)(29,31)(34,43)(39,44)(40,42)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576130);