Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357620 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1813 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,8,14)(3,5,11,10)(4,16,15,9)(6,12,7,13)(17,18,22,26)(19,27,29,40)(20,24,32,38)(21,34,28,25)(23,35,36,31)(30,41,43,48)(33,45,42,39)(37,46,47,44);(1,3,7,4,8,11,6,15)(2,9,12,10,14,16,13,5)(17,19,28,35,22,29,21,31)(18,23,34,40,26,36,25,27)(20,30,42,46,32,43,33,44)(24,37,45,48,38,47,39,41);(1,4,7,11,8,15,6,3)(2,10,12,16,14,5,13,9)(19,29)(23,36)(27,40)(30,43)(31,35)(37,47)(41,48)(44,46);(1,5)(2,11)(3,14)(4,13)(6,16)(7,9)(8,10)(12,15)(17,20)(18,24)(19,30)(21,33)(22,32)(23,37)(25,39)(26,38)(27,41)(28,42)(29,43)(31,44)(34,45)(35,46)(36,47)(40,48);(1,6,8,7)(2,12,14,13)(3,15,11,4)(5,9,10,16)(20,32)(24,38)(30,43)(33,42)(37,47)(39,45)(41,48)(44,46)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6264851);