Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 |
| prime radical | #23421 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,22)(16,29)(17,23)(18,24)(19,34)(20,25)(21,27)(26,36)(28,37)(30,43)(31,38)(32,40)(33,39)(35,41)(42,44);(1,2,3,4)(7,8)(10,12)(13,15,19,27,20,28,18,16)(14,21,24,22,25,29,34,37)(17,26,31,41,32,42,33,30)(23,35,39,36,40,43,38,44);(5,7,9,8)(6,10,11,12)(13,16,18,28,20,27,19,15)(14,22,34,21,25,37,24,29)(17,30,33,42,32,41,31,26)(23,36,38,35,40,44,39,43);(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,26)(16,30)(18,33)(19,31)(20,32)(21,35)(22,36)(24,39)(25,40)(27,41)(28,42)(29,43)(34,38)(37,44);(13,18,20,19)(14,24,25,34)(15,16,28,27)(17,31,32,33)(21,22,29,37)(23,38,40,39)(26,41,42,30)(35,44,43,36)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370410);