Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 |
| prime radical | #61414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1662 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)(17,18,24,30,23,28,22,29)(19,31,34,25,26,20,33,40)(21,27,41,45,38,46,39,44)(32,37,47,42,43,35,36,48);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,29)(21,36)(22,34)(23,26)(24,33)(27,37)(28,40)(30,31)(32,39)(35,46)(38,47)(41,43)(42,44)(45,48);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,24,40,23,31,22,25)(18,26,29,34,28,19,30,33)(21,37,41,42,38,35,39,48)(27,32,44,36,46,43,45,47);(3,9)(6,13)(10,15)(14,16)(17,21,22,39,23,38,24,41)(18,27,29,44,28,46,30,45)(19,32,33,36,26,43,34,47)(20,35,25,42,31,37,40,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,28)(19,26)(21,38)(29,30)(33,34)(35,37)(39,41)(42,48)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=f,e^2=hi,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^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104023);