Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622982 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #60648 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,17,24)(11,19,29,47)(12,18,32,44)(13,20,33,48)(14,22,38,40)(15,21,25,52)(16,39,37,23)(26,45,55,68)(27,46,53,43)(28,58,31,42)(30,41,59,65)(34,50,61,64)(35,49,54,70)(36,63,62,51)(56,69,71,67)(57,72,60,66);(2,5)(6,8)(9,11,25,53,16,28,14,12,17,29,15,27,37,31,38,32)(10,18,40,58,23,43,21,19,24,44,22,42,39,46,52,47)(13,26,54,71,36,57,34,30,33,55,35,56,62,60,61,59)(20,41,64,72,51,67,49,45,48,65,50,66,63,69,70,68);(1,3,4,7)(2,6,5,8)(9,12,14,31,37,53,25,29,17,32,38,28,16,27,15,11)(10,19,21,46,39,58,40,44,24,47,52,43,23,42,22,18)(13,30,34,60,62,71,54,55,33,59,61,57,36,56,35,26)(20,45,49,69,63,72,64,65,48,68,70,67,51,66,50,41);(9,13)(10,20)(11,26)(12,30)(14,34)(15,35)(16,36)(17,33)(18,41)(19,45)(21,49)(22,50)(23,51)(24,48)(25,54)(27,56)(28,57)(29,55)(31,60)(32,59)(37,62)(38,61)(39,63)(40,64)(42,66)(43,67)(44,65)(46,69)(47,68)(52,70)(53,71)(58,72);(13,33)(20,48)(26,55)(30,59)(34,61)(35,54)(36,62)(41,65)(45,68)(49,70)(50,64)(51,63)(56,71)(57,60)(66,72)(67,69)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,f^2=hi,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,h^a=hi,c^b=ci,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530378);