Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61197 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1741 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)(3,5)(4,6)(7,8)(9,10,15,20,16,21,14,34)(11,22,26,17,25,41,35,40)(12,27,29,44,30,43,37,18)(13,19,32,47,33,48,36,46)(23,50,54,65,52,38,53,49)(24,45,31,39,55,64,56,63)(28,57,60,67,61,68,59,42)(51,69,71,72,70,62,58,66);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,23)(13,31)(14,35)(15,26)(16,25)(18,38)(19,45)(20,41)(21,40)(22,34)(24,36)(27,49)(28,58)(29,54)(30,52)(32,55)(33,56)(37,53)(39,47)(42,66)(43,65)(44,50)(46,63)(48,64)(51,60)(57,69)(59,70)(61,71)(62,68)(67,72);(1,3,4,8)(2,5,6,7)(9,12,16,30)(10,18,21,44)(11,23,25,52)(13,28,33,61)(14,29,15,37)(17,38,40,50)(19,42,48,67)(20,43,34,27)(22,49,41,65)(24,51,55,70)(26,53,35,54)(31,58,56,71)(32,59,36,60)(39,62,63,69)(45,66,64,72)(46,57,47,68);(3,8)(5,7)(9,13)(10,19)(11,24)(12,28)(14,36)(15,32)(16,33)(17,39)(18,42)(20,47)(21,48)(22,45)(23,51)(25,55)(26,31)(27,57)(29,60)(30,61)(34,46)(35,56)(37,59)(38,62)(40,63)(41,64)(43,68)(44,67)(49,66)(50,69)(52,70)(53,58)(54,71)(65,72);(2,6)(5,7)(11,25)(17,40)(22,41)(23,52)(24,55)(26,35)(31,56)(38,50)(39,63)(45,64)(49,65)(51,70)(53,54)(58,71)(62,69)(66,72)Polycyclic presentation:
2^9|a^2=g,c^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970335);