Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366710 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,13)(4,14)(5,11)(6,10)(7,12)(8,15)(9,16)(17,18,23,27)(19,29,20,25)(21,26,36,43)(22,37,39,28)(24,33,41,31)(30,46,32,42)(34,48,47,40)(35,45,38,44);(1,3,7,8)(2,4,12,16)(5,14,6,9)(10,13,11,15)(17,19)(18,24)(20,23)(21,34)(22,33)(25,28)(26,42)(27,41)(29,37)(30,35)(31,39)(32,38)(36,47)(40,44)(43,46)(45,48);(1,4,7,16)(2,8,12,3)(5,15,6,13)(9,10,14,11)(17,20,23,19)(18,25,27,29)(21,32,36,30)(22,33,39,31)(24,37,41,28)(26,42,43,46)(34,35,47,38)(40,44,48,45);(1,4,7,16)(2,9,12,14)(3,11,8,10)(5,13,6,15)(17,21,23,36)(18,26,27,43)(19,30,20,32)(22,38,39,35)(24,40,41,48)(25,42,29,46)(28,45,37,44)(31,47,33,34);(1,5,7,6)(2,10,12,11)(3,14,8,9)(4,13,16,15)(18,27)(19,20)(26,43)(28,37)(30,32)(31,33)(34,47)(44,45)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274057);