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 | #60932 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(3,7)(6,8)(9,10,15,20,16,21,14,36)(11,22,26,17,25,43,37,42)(12,27,30,47,31,46,39,18)(13,19,34,51,35,52,38,50)(23,44,57,61,28,40,56,53)(24,54,60,41,59,48,32,66)(29,62,65,49,33,67,64,45)(55,70,68,72,58,69,63,71);(2,5)(3,7)(9,11)(10,17)(12,28)(13,32)(14,37)(15,26)(16,25)(18,44)(19,48)(20,43)(21,42)(22,36)(23,31)(24,34)(27,61)(29,63)(30,56)(33,68)(35,60)(38,59)(39,57)(40,47)(41,50)(45,71)(46,53)(49,72)(51,66)(52,54)(55,65)(58,64)(62,70)(67,69);(1,2)(3,6)(4,5)(7,8)(9,12,16,31)(10,18,21,47)(11,23,25,28)(13,33,35,29)(14,30,15,39)(17,40,42,44)(19,49,52,45)(20,46,36,27)(22,53,43,61)(24,58,59,55)(26,56,37,57)(32,63,60,68)(34,65,38,64)(41,70,66,69)(48,71,54,72)(50,67,51,62);(2,5)(6,8)(9,13,16,35)(10,19,21,52)(11,24,25,59)(12,29,31,33)(14,38,15,34)(17,41,42,66)(18,45,47,49)(20,51,36,50)(22,54,43,48)(23,55,28,58)(26,60,37,32)(27,62,46,67)(30,65,39,64)(40,69,44,70)(53,71,61,72)(56,63,57,68);(1,3,4,7)(2,6,5,8)(11,25)(17,42)(22,43)(23,28)(24,59)(26,37)(32,60)(40,44)(41,66)(48,54)(53,61)(55,58)(56,57)(63,68)(69,70)(71,72)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7055706);