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 | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2236 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)(9,10,16,21,15,22,14,20)(11,23,27,17,28,46,26,39)(12,29,35,42,32,41,31,18)(13,19,36,45,33,43,34,44)(24,49,52,47,53,37,51,65)(25,48,56,38,54,66,55,62)(30,57,60,63,58,64,59,40)(50,68,71,67,69,61,70,72);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,24)(13,25)(14,27)(15,28)(16,26)(18,37)(19,38)(20,46)(21,23)(22,39)(29,47)(30,50)(31,52)(32,53)(33,54)(34,56)(35,51)(36,55)(40,61)(41,65)(42,49)(43,62)(44,66)(45,48)(57,67)(58,69)(59,71)(60,70)(63,68)(64,72);(2,6)(3,7)(9,12,16,35,15,32,14,31)(10,18,20,41,22,42,21,29)(11,24,26,51,28,53,27,52)(13,30,36,60,33,58,34,59)(17,37,46,65,39,49,23,47)(19,40,44,64,43,63,45,57)(25,50,55,70,54,69,56,71)(38,61,66,72,62,68,48,67);(3,7)(5,8)(9,13)(10,19)(11,25)(12,30)(14,34)(15,33)(16,36)(17,38)(18,40)(20,44)(21,45)(22,43)(23,48)(24,50)(26,55)(27,56)(28,54)(29,57)(31,59)(32,58)(35,60)(37,61)(39,62)(41,64)(42,63)(46,66)(47,67)(49,68)(51,70)(52,71)(53,69)(65,72);(13,33)(19,43)(25,54)(30,58)(34,36)(38,62)(40,63)(44,45)(48,66)(50,69)(55,56)(57,64)(59,60)(61,68)(67,72)(70,71)Polycyclic presentation:
2^9|a^2=fi,c^2=ghi,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366797);