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 | |
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)(17,18,24,30)(19,31,36,44)(20,37,39,43)(21,27,32,42)(22,29,23,28)(25,34,45,35)(26,33,46,38)(40,48,41,47);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,21)(22,35)(23,34)(24,36)(26,27)(28,31)(29,44)(30,45)(32,39)(33,40)(37,48)(38,41)(42,46)(43,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,22,33,24,39,23,38)(18,26,28,43,30,46,29,37)(19,32,35,41,36,21,34,40)(25,42,31,48,45,27,44,47);(3,9)(6,13)(10,15)(14,16)(17,21,24,32)(18,27,30,42)(19,33,36,38)(20,35,39,34)(22,40,23,41)(25,43,45,37)(26,31,46,44)(28,47,29,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,38,39,33)(21,40,32,41)(25,44,45,31)(26,37,46,43)(27,47,42,48)Polycyclic presentation:
2^9|a^2=h,c^2=gi,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7081037);