Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 | #61424 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,22,29,24,28,23,41)(19,30,34,44,33,43,39,25)(20,35,37,26,32,47,38,46)(21,27,36,42,40,48,31,45);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,33)(18,25,28,44)(20,36,32,31)(21,38,40,37)(22,39,23,34)(26,45,46,42)(27,47,48,35)(29,43,41,30);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,24,32)(18,26,28,46)(19,31,33,36)(21,39,40,34)(22,37,23,38)(25,42,44,45)(27,30,48,43)(29,47,41,35);(3,10)(6,9)(11,16)(14,15)(17,21,23,31,24,40,22,36)(18,27,41,45,28,48,29,42)(19,32,39,37,33,20,34,38)(25,26,43,35,44,46,30,47);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,28)(19,33)(21,40)(26,46)(29,41)(31,36)(34,39)(35,47)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=i,d^2=fi,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,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088504);