Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,26)(13,32)(14,35)(15,21)(16,20)(18,29)(19,33)(22,49)(23,52)(24,42)(25,40)(27,38)(28,44)(30,45)(31,48)(34,47)(36,46)(37,41)(39,53)(43,51)(50,62)(54,64)(55,65)(56,63)(57,59)(58,66)(60,67)(61,68)(69,70)(71,72);(1,3,4,8)(2,5,6,7)(9,11,16,25)(10,17,20,40)(12,27,31,56)(13,23,34,55)(14,24,15,37)(18,43,45,49)(19,39,46,64)(21,41,35,42)(22,29,51,30)(26,38,48,63)(28,58,60,70)(32,52,47,65)(33,53,36,54)(44,66,67,69)(50,61,71,59)(57,62,68,72);(9,12,16,31)(10,18,20,45)(11,22,25,51)(13,28,34,60)(14,30,15,29)(17,38,40,63)(19,44,46,67)(21,48,35,26)(23,50,55,71)(24,56,37,27)(32,57,47,68)(33,61,36,59)(39,62,64,72)(41,43,42,49)(52,69,65,66)(53,58,54,70);(2,6)(3,8)(9,13)(10,19)(11,23)(12,28)(14,36)(15,33)(16,34)(17,39)(18,44)(20,46)(21,47)(22,50)(24,54)(25,55)(26,57)(27,58)(29,61)(30,59)(31,60)(32,35)(37,53)(38,62)(40,64)(41,65)(42,52)(43,66)(45,67)(48,68)(49,69)(51,71)(56,70)(63,72);(10,20)(17,40)(18,45)(19,46)(21,35)(26,48)(32,47)(38,63)(39,64)(41,42)(43,49)(44,67)(52,65)(57,68)(62,72)(66,69)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408337);