Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624824 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1287 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,27)(13,19)(14,33)(15,36)(16,21)(18,30)(20,34)(22,46)(23,38)(24,39)(25,53)(26,41)(28,45)(29,55)(31,61)(32,57)(35,42)(37,50)(40,52)(43,58)(44,60)(47,66)(48,67)(49,68)(51,69)(54,62)(56,59)(63,70)(64,71)(65,72);(1,3)(2,5)(4,8)(6,7)(9,11,16,26)(10,17,21,41)(12,22,32,51)(13,23,15,24)(14,25,35,54)(18,37,44,65)(19,38,36,39)(20,40,45,66)(27,46,57,69)(28,47,34,52)(29,48,31,49)(30,50,60,72)(33,53,42,62)(43,63,56,64)(55,67,61,68)(58,70,59,71);(9,12,28,58,15,31,14,30,16,32,34,59,13,29,35,60)(10,18,42,61,19,43,20,27,21,44,33,55,36,56,45,57)(11,22,47,70,24,49,25,50,26,51,52,71,23,48,54,72)(17,37,62,68,38,63,40,46,41,65,53,67,39,64,66,69);(3,8)(5,7)(9,13,16,15)(10,19,21,36)(11,23,26,24)(12,29,32,31)(14,28,35,34)(17,38,41,39)(18,43,44,56)(20,33,45,42)(22,48,51,49)(25,47,54,52)(27,55,57,61)(30,58,60,59)(37,63,65,64)(40,53,66,62)(46,67,69,68)(50,70,72,71);(2,6)(5,7)(9,13,16,15)(10,19,21,36)(11,24,26,23)(12,29,32,31)(14,28,35,34)(17,39,41,38)(18,43,44,56)(20,33,45,42)(22,49,51,48)(25,52,54,47)(27,55,57,61)(30,58,60,59)(37,64,65,63)(40,62,66,53)(46,68,69,67)(50,71,72,70)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532031);