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 | |
1662 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)(17,18,23,30,24,19,22,27)(20,31,35,25,33,44,34,28)(21,26,39,36,40,29,38,43)(32,45,47,41,37,48,46,42);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,18)(20,25,33,28)(21,36,40,43)(22,27,23,30)(26,38,29,39)(31,34,44,35)(32,45,37,48)(41,47,42,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,23,35,24,33,22,34)(18,25,27,31,19,28,30,44)(21,37,39,46,40,32,38,47)(26,42,43,48,29,41,36,45);(3,10)(6,9)(11,16)(14,15)(17,21,23,39,24,40,22,38)(18,26,30,36,19,29,27,43)(20,32,35,47,33,37,34,46)(25,41,44,48,28,42,31,45);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,19)(20,33)(26,29)(27,30)(32,37)(34,35)(36,43)(46,47)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=g,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,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093928);