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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,22,28,24,29,23,41)(19,30,34,44,35,43,33,25)(20,36,37,26,32,45,38,46)(21,27,31,47,40,48,39,42);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,35)(18,25,29,44)(20,31,32,39)(21,38,40,37)(22,33,23,34)(26,42,46,47)(27,45,48,36)(28,43,41,30);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,24,32)(18,26,29,46)(19,31,35,39)(21,34,40,33)(22,37,23,38)(25,42,44,47)(27,43,48,30)(28,45,41,36);(3,10)(6,9)(11,16)(14,15)(17,21,23,39,24,40,22,31)(18,27,41,42,29,48,28,47)(19,32,33,37,35,20,34,38)(25,26,43,36,44,46,30,45);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,24,23)(18,28,29,41)(19,33,35,34)(20,37,32,38)(21,39,40,31)(25,43,44,30)(26,45,46,36)(27,42,48,47)Polycyclic presentation:
2^9|a^2=f,b^2=hi,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, 7088512);