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 | #532 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1582 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)(17,18,22,27,23,26,39,28)(19,29,32,42,31,46,33,24)(20,34,21,25,36,44,38,45)(30,43,37,48,35,40,47,41);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,23,31)(18,24,26,42)(20,35,36,30)(21,37,38,47)(22,33,39,32)(25,43,45,40)(27,46,28,29)(34,48,44,41);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20,22,21,23,36,39,38)(18,25,28,34,26,45,27,44)(19,30,33,47,31,35,32,37)(24,40,29,41,42,43,46,48);(3,10)(6,9)(11,16)(14,15)(17,21,22,36,23,38,39,20)(18,25,27,44,26,45,28,34)(19,30,32,37,31,35,33,47)(24,41,29,43,42,48,46,40);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(18,26)(19,31)(20,36)(21,38)(27,28)(32,33)(40,43)(41,48)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=g,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=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093945);