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 | #3036 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1593 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)(17,18,22,28,24,29,23,27)(19,20,31,25,33,36,32,35)(21,26,39,45,41,46,40,44)(30,34,47,42,43,38,37,48);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,24,33)(18,25,29,35)(20,27,36,28)(21,37,41,47)(22,32,23,31)(26,34,46,38)(30,39,43,40)(42,45,48,44);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,23,35,24,36,22,25)(18,19,28,31,29,33,27,32)(21,38,40,42,41,34,39,48)(26,43,45,37,46,30,44,47);(3,10)(6,9)(11,16)(14,15)(17,21,22,39,24,41,23,40)(18,26,28,45,29,46,27,44)(19,30,31,47,33,43,32,37)(20,34,25,42,36,38,35,48);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,24,23)(18,27,29,28)(19,31,33,32)(20,35,36,25)(21,39,41,40)(26,44,46,45)(30,47,43,37)(34,48,38,42)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=g,d^2=fi,e^2=h,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, 7093924);