Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624184 |
| 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 | |
1286 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)(9,10,16,22)(11,17,25,40)(12,27,30,59)(13,21,15,19)(14,33,36,46)(18,42,44,29)(20,32,48,35)(23,50,52,71)(24,45,31,39)(26,57,56,69)(28,60,37,47)(34,49,62,43)(38,63,65,53)(41,61,67,55)(51,68,54,70)(58,64,72,66);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,23)(13,31)(14,26)(15,24)(16,25)(18,38)(19,45)(20,41)(21,39)(22,40)(27,50)(28,54)(29,53)(30,52)(32,61)(33,57)(34,58)(35,55)(36,56)(37,51)(42,63)(43,66)(44,65)(46,69)(47,68)(48,67)(49,64)(59,71)(60,70)(62,72);(9,12,16,30)(10,18,22,44)(11,23,25,52)(13,28,15,37)(14,34,36,62)(17,38,40,65)(19,43,21,49)(20,47,48,60)(24,51,31,54)(26,58,56,72)(27,46,59,33)(29,32,42,35)(39,64,45,66)(41,68,67,70)(50,69,71,57)(53,61,63,55);(2,5)(6,8)(9,13,16,15)(10,19,22,21)(11,24,25,31)(12,28,30,37)(14,32,36,35)(17,39,40,45)(18,43,44,49)(20,46,48,33)(23,51,52,54)(26,55,56,61)(27,47,59,60)(29,34,42,62)(38,64,65,66)(41,57,67,69)(50,70,71,68)(53,72,63,58);(1,3,4,7)(2,6,5,8)(11,25)(17,40)(23,52)(24,31)(26,56)(38,65)(39,45)(41,67)(50,71)(51,54)(53,63)(55,61)(57,69)(58,72)(64,66)(68,70)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531837);