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 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,9,11,15)(4,13,14,8)(6,10,16,12)(17,18)(19,24)(20,32)(21,26)(22,35)(23,27)(25,33)(28,42)(29,38)(30,39)(31,45)(34,40)(36,41)(37,43)(44,47)(46,48);(1,3,5,11)(2,6,7,16)(4,10,14,12)(8,15,13,9)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,37)(26,38)(27,39)(32,42)(33,43)(34,44)(35,45)(36,46)(40,47)(41,48);(17,20)(18,25)(19,28)(21,33)(22,36)(23,34)(24,37)(26,40)(27,41)(29,43)(30,44)(31,46)(32,35)(38,47)(39,48)(42,45);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(17,21,23,22)(18,26,27,35)(19,29,30,31)(20,33,34,36)(24,38,39,45)(25,40,41,32)(28,43,44,46)(37,47,48,42);(2,7)(3,11)(8,13)(10,12)(19,30)(24,39)(28,44)(29,31)(37,48)(38,45)(42,47)(43,46)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^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, 6271577);