Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355448 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60940 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,10)(4,7)(9,14)(11,12)(13,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,41)(30,42)(31,38)(33,39)(34,40)(35,46)(37,44)(43,47)(45,48);(1,2,7,14,5,9,4,6)(3,8,10,16,11,15,12,13)(17,19,21,31,23,30,22,32)(18,24,36,42,27,38,26,28)(20,29,34,43,33,45,35,44)(25,37,46,48,39,47,40,41);(1,3,5,11)(2,8,9,15)(4,12,7,10)(6,13,14,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(30,45)(31,43)(32,44)(36,46)(38,47)(42,48);(3,11)(8,15)(10,12)(13,16)(19,30)(20,33)(24,38)(25,39)(28,42)(31,32)(34,35)(40,46);(1,4,5,7)(2,6,9,14)(3,12,11,10)(8,13,15,16)(17,21,23,22)(18,26,27,36)(19,31,30,32)(20,34,33,35)(24,28,38,42)(25,40,39,46)(29,43,45,44)(37,41,47,48)Polycyclic presentation:
2^9|b^2=fh,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=dhi,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262653);