Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| 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 |
| prime radical | #61473 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1257 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)(6,8)(9,10,15,18)(11,16,22,39)(12,25,28,59)(13,29,33,58)(14,34,31,20)(17,40,41,26)(19,44,43,32)(21,47,48,70)(23,53,51,69)(24,56,52,38)(27,60,35,45)(30,46,61,42)(36,62,64,49)(37,65,67,55)(50,71,57,66)(54,68,72,63);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,23)(14,24)(15,22)(17,36)(18,39)(19,37)(20,38)(25,47)(26,49)(27,50)(28,48)(29,53)(30,54)(31,52)(32,55)(33,51)(34,56)(35,57)(40,62)(41,64)(42,63)(43,67)(44,65)(45,66)(46,68)(58,69)(59,70)(60,71)(61,72);(9,12,15,28)(10,17,18,41)(11,21,22,48)(13,30,33,61)(14,35,31,27)(16,36,39,64)(19,45,43,60)(20,46,34,42)(23,54,51,72)(24,57,52,50)(25,58,59,29)(26,44,40,32)(37,66,67,71)(38,68,56,63)(47,69,70,53)(49,65,62,55);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(10,18)(11,22)(17,41)(19,43)(20,34)(21,48)(23,51)(24,52)(25,59)(29,58)(42,46)(45,60)(49,62)(50,57)(54,72)(55,65)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531304);