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 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10)(11,17)(12,27)(13,19)(14,34)(15,36)(16,21)(18,30)(20,28)(22,44)(23,38)(24,39)(25,52)(26,53)(29,54)(31,60)(32,56)(33,43)(35,61)(37,48)(40,45)(41,58)(42,59)(46,65)(47,66)(49,71)(50,64)(51,72)(55,57)(62,69)(63,70)(67,68);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,22)(13,23)(14,25)(15,26)(16,24)(18,37)(19,38)(20,40)(21,39)(27,44)(28,45)(29,46)(30,48)(31,49)(32,47)(33,50)(34,52)(35,51)(36,53)(41,62)(42,63)(43,64)(54,65)(55,67)(56,66)(57,68)(58,69)(59,70)(60,71)(61,72);(9,12,28,57,15,31,35,59,16,32,33,58,13,29,14,30)(10,18,34,54,19,41,43,56,21,42,61,60,36,55,20,27)(11,22,45,68,26,49,51,70,24,47,50,69,23,46,25,48)(17,37,52,65,38,62,64,66,39,63,72,71,53,67,40,44);(2,5)(6,8)(9,13,16,15)(10,19,21,36)(11,23,24,26)(12,29,32,31)(14,33,35,28)(17,38,39,53)(18,41,42,55)(20,34,43,61)(22,46,47,49)(25,50,51,45)(27,54,56,60)(30,58,59,57)(37,62,63,67)(40,52,64,72)(44,65,66,71)(48,69,70,68);(1,3,4,7)(2,6,5,8)(11,24)(17,39)(22,47)(23,26)(25,51)(37,63)(38,53)(40,64)(44,66)(45,50)(46,49)(48,70)(52,72)(62,67)(65,71)(68,69)Polycyclic presentation:
2^9|c^2=gh,d^2=i,e^2=f,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, 7532188);