Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| 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 | #61332 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1615 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)(17,18,22,28,24,27,23,39)(19,29,33,43,32,42,38,25)(20,34,36,45,35,47,21,26)(30,40,46,41,31,44,37,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,32)(18,25,27,43)(20,31,35,30)(21,37,36,46)(22,38,23,33)(26,44,45,40)(28,42,39,29)(34,41,47,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,23,21,24,35,22,36)(18,26,28,34,27,45,39,47)(19,30,33,46,32,31,38,37)(25,40,42,48,43,44,29,41);(3,10)(6,9)(11,16)(14,15)(17,21,23,35,24,36,22,20)(18,26,39,47,27,45,28,34)(19,31,38,46,32,30,33,37)(25,41,42,40,43,48,29,44);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,27)(19,32)(20,35)(21,36)(28,39)(33,38)(40,44)(41,48)Polycyclic presentation:
2^9|a^2=f,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=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093949);