Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #332583 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3497 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,7)(4,8)(5,6)(9,10,16,22)(11,23,27,42)(12,18,32,46)(13,33,36,49)(14,21,15,20)(17,28,41,26)(19,37,48,35)(24,52,54,43)(25,55,56,40)(29,59,58,39)(30,61,64,68)(31,50,38,45)(34,47,57,51)(44,63,69,62)(53,66,71,72)(60,70,65,67);(1,3,4,5)(2,6,8,7)(9,11)(10,17)(12,29)(13,34)(14,28)(15,26)(16,27)(18,43)(19,47)(20,42)(21,23)(22,41)(24,31)(25,35)(30,53)(32,58)(33,40)(36,57)(37,56)(38,54)(39,45)(44,66)(46,52)(48,51)(49,55)(50,59)(60,62)(61,67)(63,65)(64,71)(68,70)(69,72);(2,6)(3,5)(7,8)(9,12)(10,18)(11,24)(13,30)(14,31)(15,38)(16,32)(17,39)(19,44)(20,45)(21,50)(22,46)(23,43)(25,53)(26,29)(27,54)(28,58)(33,61)(34,65)(35,62)(36,64)(37,63)(40,66)(41,59)(42,52)(47,70)(48,69)(49,68)(51,67)(55,72)(56,71)(57,60);(2,7)(3,5)(6,8)(9,13,16,36)(10,19,22,48)(11,25,27,56)(12,30,32,64)(14,37,15,35)(17,40,41,55)(18,44,46,69)(20,49,21,33)(23,51,42,47)(24,53,54,71)(26,57,28,34)(29,60,58,65)(31,63,38,62)(39,66,59,72)(43,67,52,70)(45,68,50,61)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 239828);