Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350525 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62992 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,8)(3,10,12,16)(4,9,5,13)(7,14,15,11)(17,18)(19,30)(20,35)(21,27)(22,28)(23,29)(24,33)(25,36)(26,38)(31,44)(32,42)(34,41)(37,43)(39,47)(40,46)(45,48);(1,3,6,12)(2,7,8,15)(4,11,5,14)(9,16,13,10)(17,19,23,34)(18,24,29,42)(20,31,38,47)(21,32,22,33)(25,39,43,44)(26,40,35,48)(27,30,28,41)(36,45,37,46);(17,20)(18,25)(19,31)(21,36)(22,37)(23,38)(24,39)(26,27)(28,35)(29,43)(30,40)(32,45)(33,46)(34,47)(41,48)(42,44);(2,8)(7,15)(9,13)(10,16)(17,20)(18,26)(19,31)(21,37)(22,36)(23,38)(24,40)(25,27)(28,43)(29,35)(30,39)(32,46)(33,45)(34,47)(41,44)(42,48);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,21,23,22)(18,27,29,28)(19,32,34,33)(20,36,38,37)(24,30,42,41)(25,26,43,35)(31,45,47,46)(39,40,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257764);