Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364250 |
| 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 | |
1495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,7,10)(3,12,14,16)(4,11,6,8)(5,9,15,13)(17,18)(19,24)(20,31)(21,26)(22,35)(23,27)(25,32)(28,43)(29,38)(30,40)(33,41)(34,39)(36,42)(37,45)(44,47)(46,48);(1,3,7,14)(2,5,10,15)(4,13,6,9)(8,12,11,16)(17,19,23,30)(18,24,27,40)(20,28,33,46)(21,34,22,29)(25,37,42,47)(26,39,35,38)(31,43,41,48)(32,45,36,44);(1,4,7,6)(2,8,10,11)(3,13,14,9)(5,12,15,16)(17,20)(18,25)(19,28)(21,32)(22,36)(23,33)(24,37)(26,41)(27,42)(29,44)(30,46)(31,35)(34,45)(38,43)(39,48)(40,47);(1,5,7,15)(2,9,10,13)(3,11,14,8)(4,12,6,16);(2,10)(3,14)(8,11)(9,13)(17,21,23,22)(18,26,27,35)(19,29,30,34)(20,32,33,36)(24,38,40,39)(25,41,42,31)(28,44,46,45)(37,43,47,48)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6271781);