Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624170 |
| 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 | |
1305 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)(9,10,16,21)(11,17,24,40)(12,26,29,59)(13,19,15,36)(14,32,35,61)(18,43,46,28)(20,31,48,34)(22,49,51,70)(23,39,30,42)(25,56,55,66)(27,58,37,47)(33,44,62,45)(38,63,65,52)(41,60,67,54)(50,68,53,72)(57,69,71,64);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,22)(13,30)(14,25)(15,23)(16,24)(18,38)(19,42)(20,41)(21,40)(26,49)(27,53)(28,52)(29,51)(31,60)(32,56)(33,57)(34,54)(35,55)(36,39)(37,50)(43,63)(44,69)(45,64)(46,65)(47,68)(48,67)(58,72)(59,70)(61,66)(62,71);(9,12)(10,18)(11,22)(13,27)(14,33)(15,37)(16,29)(17,38)(19,44)(20,47)(21,46)(23,50)(24,51)(25,57)(26,32)(28,34)(30,53)(31,43)(35,62)(36,45)(39,64)(40,65)(41,68)(42,69)(48,58)(49,56)(52,54)(55,71)(59,61)(60,63)(66,70)(67,72);(2,5)(6,8)(9,13,16,15)(10,19,21,36)(11,23,24,30)(12,27,29,37)(14,31,35,34)(17,39,40,42)(18,44,46,45)(20,32,48,61)(22,50,51,53)(25,54,55,60)(26,58,59,47)(28,33,43,62)(38,64,65,69)(41,66,67,56)(49,68,70,72)(52,71,63,57);(1,3,4,7)(2,6,5,8)(11,24)(17,40)(22,51)(23,30)(25,55)(38,65)(39,42)(41,67)(49,70)(50,53)(52,63)(54,60)(56,66)(57,71)(64,69)(68,72)Polycyclic presentation:
2^9|a^2=i,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=dfi,e^b=ei,g^c=gh,h^c=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531815);