Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470818 |
| 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 | |
1914 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)(11,21,25,44)(12,27,30,17)(13,18,31,40)(14,33,22,19)(16,34,37,26)(23,47,49,42)(24,43,52,65)(28,54,56,38)(29,39,45,55)(32,58,46,41)(35,50,53,59)(36,60,62,51)(48,67,69,64)(57,63,66,71)(61,68,70,72);(1,3)(2,5)(4,7)(6,8)(9,11,22,34,15,25,14,26)(10,16,33,44,20,37,19,21)(12,23,45,59,30,49,29,50)(13,24,46,60,31,52,32,51)(17,35,55,47,27,53,39,42)(18,36,58,65,40,62,41,43)(28,48,66,72,56,69,57,68)(38,61,71,67,54,70,63,64);(3,7)(5,8)(9,12,14,29,15,30,22,45)(10,17,19,39,20,27,33,55)(11,23,26,50,25,49,34,59)(13,28,32,57,31,56,46,66)(16,35,21,42,37,53,44,47)(18,38,41,63,40,54,58,71)(24,48,51,68,52,69,60,72)(36,61,43,64,62,70,65,67);(3,7)(5,8)(9,13)(10,18)(11,24)(12,28)(14,32)(15,31)(16,36)(17,38)(19,41)(20,40)(21,43)(22,46)(23,48)(25,52)(26,51)(27,54)(29,57)(30,56)(33,58)(34,60)(35,61)(37,62)(39,63)(42,64)(44,65)(45,66)(47,67)(49,69)(50,68)(53,70)(55,71)(59,72);(2,6)(5,8)(11,25)(13,31)(16,37)(18,40)(21,44)(23,49)(26,34)(28,56)(32,46)(35,53)(38,54)(41,58)(42,47)(50,59)(57,66)(63,71)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378023);