Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357957 |
| 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 | |
1815 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)(3,5)(4,6)(7,10)(8,14)(9,11)(12,13)(15,16)(17,18,22,24)(19,27,31,40)(20,25,32,38)(21,34,28,26)(23,35,36,30)(29,41,43,48)(33,45,42,39)(37,46,47,44);(1,3,11,15,7,13,6,14)(2,8,4,12,10,16,9,5)(17,19,28,35,22,31,21,30)(18,23,34,40,24,36,26,27)(20,29,42,46,32,43,33,44)(25,37,45,48,38,47,39,41);(1,4,11,10,7,9,6,2)(3,12,15,16,13,5,14,8)(18,24)(23,36)(25,38)(26,34)(27,40)(37,47)(39,45)(41,48);(1,5)(2,3)(4,15)(6,16)(7,12)(8,11)(9,14)(10,13)(17,20)(18,25)(19,29)(21,33)(22,32)(23,37)(24,38)(26,39)(27,41)(28,42)(30,44)(31,43)(34,45)(35,46)(36,47)(40,48);(3,13)(5,12)(8,16)(14,15)(20,32)(25,38)(29,43)(33,42)(37,47)(39,45)(41,48)(44,46)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=di,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265192);