Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| 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 | |
1664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,22,27,23,29,39,28)(19,30,32,42,34,45,33,24)(20,35,37,25,36,43,21,26)(31,44,38,48,46,40,47,41);(3,9)(4,12)(6,13)(7,11)(17,19)(18,24)(20,31)(21,38)(22,33)(23,34)(25,40)(26,44)(27,45)(28,30)(29,42)(32,39)(35,41)(36,46)(37,47)(43,48);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,22,37,23,36,39,21)(18,25,28,35,29,26,27,43)(19,31,33,47,34,46,32,38)(24,40,30,41,42,44,45,48);(3,9)(6,13)(10,15)(14,16)(17,21,39,36,23,37,22,20)(18,26,28,43,29,25,27,35)(19,31,33,47,34,46,32,38)(24,41,45,40,42,48,30,44);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(20,36)(21,37)(25,26)(31,46)(35,43)(38,47)(40,44)(41,48)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104031);