Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443050 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3455 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,22)(12,18)(13,19)(14,37)(15,20)(16,21)(17,25)(23,51)(24,52)(26,41)(27,39)(28,44)(29,61)(30,45)(31,46)(32,40)(33,48)(34,49)(35,64)(36,50)(38,42)(43,54)(47,56)(53,71)(55,69)(57,65)(58,66)(59,72)(60,70)(62,67)(63,68);(9,11,14,25,16,26,15,38)(10,17,20,22,21,42,37,41)(12,27,29,54,31,58,30,23)(13,32,35,56,34,63,36,24)(18,43,45,39,46,51,61,66)(19,47,50,40,49,52,64,68)(28,55,59,62,33,53,60,57)(44,67,70,69,48,65,72,71);(2,5)(6,8)(9,12,14,29,16,31,15,30)(10,18,37,61,21,46,20,45)(11,23,38,58,26,54,25,27)(13,33,35,60,34,28,36,59)(17,39,22,51,42,66,41,43)(19,48,64,70,49,44,50,72)(24,55,63,57,56,53,32,62)(40,67,52,69,68,65,47,71);(9,13,14,35,16,34,15,36)(10,19,37,64,21,49,20,50)(11,24,38,63,26,56,25,32)(12,28,29,59,31,33,30,60)(17,40,22,52,42,68,41,47)(18,44,61,72,46,48,45,70)(23,53,58,62,54,55,27,57)(39,65,51,71,66,67,43,69);(1,3,4,7)(2,6,5,8)(13,34)(19,49)(24,56)(28,33)(32,63)(35,36)(40,68)(44,48)(47,52)(50,64)(53,55)(57,62)(59,60)(65,67)(69,71)(70,72)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375055);