Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(7,11)(13,14,19,23)(15,20,18,33)(16,26,29,43)(17,22,32,39)(21,24,36,28)(25,34,30,37)(27,42,31,44)(35,40,38,41);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,18)(14,20,23,33)(16,24,29,28)(17,30,32,25)(21,26,36,43)(22,37,39,34)(27,41,31,40)(35,44,38,42);(2,4)(13,16,18,28,19,29,15,24)(14,21,20,26,23,36,33,43)(17,31,30,40,32,27,25,41)(22,38,34,44,39,35,37,42);(5,7,8,11)(6,10,9,12)(13,17,19,32)(14,22,23,39)(15,25,18,30)(16,27,29,31)(20,34,33,37)(21,35,36,38)(24,40,28,41)(26,42,43,44);(2,4)(6,9)(10,12)(17,32)(22,39)(25,30)(27,31)(34,37)(35,38)(40,41)(42,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590934);