Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438886 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,7)(3,6,4,8)(9,10,16,37,15,21,14,20)(11,22,25,17,27,41,26,42)(12,28,31,46,33,18,32,45)(13,19,38,64,36,49,35,48)(23,52,57,43,29,50,54,39)(24,51,55,40,34,63,56,47)(30,58,60,68,62,44,61,67)(53,70,72,66,59,69,71,65);(1,3,5,4)(2,6,7,8)(9,11)(10,17)(12,29)(13,34)(14,25)(15,27)(16,26)(18,43)(19,47)(20,41)(21,42)(22,37)(23,33)(24,36)(28,39)(30,59)(31,57)(32,54)(35,56)(38,55)(40,49)(44,66)(45,52)(46,50)(48,51)(53,62)(58,65)(60,72)(61,71)(63,64)(67,70)(68,69);(1,4,5,3)(2,6,7,8)(9,12,16,31,15,33,14,32)(10,18,37,45,21,28,20,46)(11,23,25,57,27,29,26,54)(13,30,38,60,36,62,35,61)(17,39,41,52,42,43,22,50)(19,44,64,67,49,58,48,68)(24,53,55,72,34,59,56,71)(40,65,63,70,47,66,51,69);(9,13,14,35,15,36,16,38)(10,19,20,48,21,49,37,64)(11,24,26,56,27,34,25,55)(12,30,32,61,33,62,31,60)(17,40,22,51,42,47,41,63)(18,44,46,68,28,58,45,67)(23,53,54,71,29,59,57,72)(39,65,50,69,43,66,52,70);(9,14,15,16)(10,20,21,37)(11,25,27,26)(12,31,33,32)(13,35,36,38)(17,41,42,22)(18,45,28,46)(19,48,49,64)(23,54,29,57)(24,55,34,56)(30,60,62,61)(39,50,43,52)(40,63,47,51)(44,67,58,68)(53,71,59,72)(65,69,66,70)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355279);