Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350061 |
| 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 | #63018 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,10)(3,8,12,15)(4,9,5,14)(7,13,16,11)(17,18)(19,24)(20,31)(21,27)(22,34)(23,26)(25,32)(28,43)(29,39)(30,38)(33,42)(35,40)(36,41)(37,46)(44,48)(45,47);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16)(17,19,23,30)(18,24,26,38)(20,28,33,45)(21,29,22,35)(25,37,41,48)(27,39,34,40)(31,47,42,43)(32,44,36,46);(17,20)(18,25)(19,28)(21,32)(22,36)(23,33)(24,37)(26,41)(27,42)(29,44)(30,45)(31,34)(35,46)(38,48)(39,43)(40,47);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(18,26)(24,38)(25,41)(27,34)(31,42)(37,48)(39,40)(43,47);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(17,21,23,22)(18,27,26,34)(19,29,30,35)(20,32,33,36)(24,39,38,40)(25,42,41,31)(28,44,45,46)(37,43,48,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6257514);