Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,27)(21,33)(26,34)(28,36)(29,35)(31,37)(32,38)(39,42)(40,43)(41,44);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,27,19,29,18,28)(14,20,23,25,24,36,22,35)(16,26,31,40,32,41,33,42)(21,34,38,39,37,44,30,43);(7,11)(9,12)(13,16,18,33,19,32,17,31)(14,21,22,30,24,37,23,38)(15,26,28,42,29,41,27,40)(20,34,35,43,36,44,25,39);(7,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,32,33)(20,35,36,25)(21,30,37,38)(26,40,41,42)(34,43,44,39);(6,10)(9,12)(16,32)(21,37)(26,41)(30,38)(31,33)(34,44)(39,43)(40,42)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^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,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380581);