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 | |
1291 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)(14,34)(15,36)(16,21)(18,30)(20,28)(22,45)(23,38)(24,39)(25,52)(26,41)(29,54)(31,60)(32,56)(33,44)(35,61)(37,49)(40,46)(42,58)(43,59)(47,67)(48,68)(50,69)(51,66)(53,65)(55,57)(62,71)(63,70)(64,72);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,23)(14,25)(15,24)(16,26)(18,37)(19,38)(20,40)(21,41)(27,45)(28,46)(29,47)(30,49)(31,48)(32,50)(33,51)(34,52)(35,53)(36,39)(42,62)(43,64)(44,66)(54,67)(55,63)(56,69)(57,70)(58,71)(59,72)(60,68)(61,65);(9,12,28,57,15,31,35,59,16,32,33,58,13,29,14,30)(10,18,34,54,19,42,44,56,21,43,61,60,36,55,20,27)(11,22,46,70,24,48,53,72,26,50,51,71,23,47,25,49)(17,37,52,67,38,62,66,69,41,64,65,68,39,63,40,45);(3,8)(5,7)(9,13,16,15)(10,19,21,36)(11,23,26,24)(12,29,32,31)(14,33,35,28)(17,38,41,39)(18,42,43,55)(20,34,44,61)(22,47,50,48)(25,51,53,46)(27,54,56,60)(30,58,59,57)(37,62,64,63)(40,52,66,65)(45,67,69,68)(49,71,72,70);(2,6)(5,7)(9,13,16,15)(10,19,21,36)(11,24,26,23)(12,29,32,31)(14,33,35,28)(17,39,41,38)(18,42,43,55)(20,34,44,61)(22,48,50,47)(25,46,53,51)(27,54,56,60)(30,58,59,57)(37,63,64,62)(40,65,66,52)(45,68,69,67)(49,70,72,71)Polycyclic presentation:
2^9|c^2=gh,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, 7532028);