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 |
| prime radical | #24180 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1617 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)(17,18,22,28,23,27,35,29)(19,21,26,40,31,41,32,24)(20,33,38,25,37,45,48,46)(30,43,47,36,34,42,44,39);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,23,31)(18,24,27,40)(20,34,37,30)(21,28,41,29)(22,32,35,26)(25,43,46,42)(33,36,45,39)(38,47,48,44);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,35,48,23,37,22,38)(18,25,28,45,27,46,29,33)(19,30,26,47,31,34,32,44)(21,39,24,42,41,36,40,43);(3,10)(6,9)(11,16)(14,15)(17,21,22,40,23,41,35,24)(18,26,28,31,27,32,29,19)(20,36,38,42,37,39,48,43)(25,44,45,30,46,47,33,34);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,27)(19,31)(20,37)(26,32)(28,29)(36,39)(38,48)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=gi,d^2=f,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, 7093926);