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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1666 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)(17,18,23,39,24,28,22,29)(19,21,27,25,31,42,32,41)(20,33,38,26,36,48,37,46)(30,43,47,35,34,44,45,40);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,34)(21,39)(22,27)(23,32)(24,31)(26,44)(28,41)(29,42)(30,36)(33,40)(35,48)(37,45)(38,47)(43,46);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,23,38,24,36,22,37)(18,26,29,33,28,46,39,48)(19,30,32,45,31,34,27,47)(21,40,25,43,42,35,41,44);(3,9)(6,13)(10,15)(14,16)(17,21,22,41,24,42,23,25)(18,27,29,19,28,32,39,31)(20,35,37,43,36,40,38,44)(26,45,33,34,46,47,48,30);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,28)(19,31)(20,36)(27,32)(29,39)(35,40)(37,38)(43,44)Polycyclic presentation:
2^9|a^2=fh,c^2=ghi,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^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104017);