Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45094 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1836 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)(5,6)(7,9)(8,10)(11,12)(13,14,20,36,19,25,18,24)(15,26,17,21,28,41,29,23)(16,30,37,40,33,22,34,44)(27,43,35,39,31,42,32,38);(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,21,25,23)(16,31,33,27)(17,20,29,18)(22,39,30,38)(24,41,36,26)(32,37,35,34)(40,42,44,43);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,33)(14,22,25,30)(15,27,28,31)(17,35,29,32)(18,34,20,37)(21,38,23,39)(24,40,36,44)(26,42,41,43);(7,12)(9,11)(13,17,18,15,19,29,20,28)(14,23,24,41,25,21,36,26)(16,32,34,31,33,35,37,27)(22,38,40,42,30,39,44,43);(6,10)(9,11)(15,28)(16,33)(17,29)(21,23)(22,30)(26,41)(34,37)(40,44)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6972371);