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 | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10,14,20)(11,22,25,17)(12,28,30,64)(13,19,16,37)(15,34,33,62)(18,42,44,31)(21,32,46,36)(23,51,53,61)(24,49,27,40)(26,57,56,66)(29,63,38,47)(35,43,65,45)(39,54,48,70)(41,59,50,55)(52,69,60,71)(58,68,72,67);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,23)(13,24)(14,25)(15,26)(16,27)(18,39)(19,40)(20,22)(21,41)(28,61)(29,52)(30,53)(31,54)(32,55)(33,56)(34,66)(35,58)(36,59)(37,49)(38,60)(42,70)(43,67)(44,48)(45,68)(46,50)(47,69)(51,64)(57,62)(63,71)(65,72);(9,12,14,30)(10,18,20,44)(11,23,25,53)(13,29,16,38)(15,35,33,65)(17,39,22,48)(19,43,37,45)(21,47,46,63)(24,52,27,60)(26,58,56,72)(28,62,64,34)(31,32,42,36)(40,67,49,68)(41,69,50,71)(51,66,61,57)(54,55,70,59);(2,5)(6,8)(9,13,14,16)(10,19,20,37)(11,24,25,27)(12,29,30,38)(15,32,33,36)(17,40,22,49)(18,43,44,45)(21,34,46,62)(23,52,53,60)(26,55,56,59)(28,63,64,47)(31,35,42,65)(39,67,48,68)(41,66,50,57)(51,69,61,71)(54,58,70,72);(1,3,4,7)(2,6,5,8)(10,20)(11,25)(18,44)(19,37)(21,46)(23,53)(24,27)(26,56)(28,64)(34,62)(43,45)(47,63)(52,60)(54,70)(55,59)(58,72)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=fi,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, 7531827);