Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18)(19,31)(20,37)(21,27)(22,29)(23,28)(24,30)(25,35)(26,39)(32,44)(33,47)(34,43)(36,42)(38,41)(40,45)(46,48);(3,9)(4,12)(6,13)(7,11)(17,19,22,35,24,34,23,36)(18,25,28,31,30,42,29,43)(20,38,39,33,21,32,40,48)(26,44,37,47,45,41,27,46);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,26)(19,32)(21,24)(22,39)(23,40)(25,41)(27,29)(28,37)(30,45)(31,46)(33,36)(34,38)(35,48)(42,44)(43,47);(3,9)(6,13)(10,15)(14,16)(17,21,24,20)(18,27,30,37)(19,33,34,48)(22,39,23,40)(25,41,42,44)(26,28,45,29)(31,47,43,46)(32,35,38,36);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(19,34)(25,42)(31,43)(32,38)(33,48)(35,36)(41,44)(46,47)Polycyclic presentation:
2^9|a^2=hi,b^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7074350);