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 | #61325 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1613 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)(17,18,21,25,22,27,29,26)(19,28,31,23,32,37,42,38)(20,24,35,33,34,39,43,40)(30,41,46,36,45,44,48,47);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,18)(19,23)(20,33)(21,26)(22,27)(24,35)(25,29)(28,31)(30,44)(32,38)(34,40)(36,46)(37,42)(39,43)(41,45)(47,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,19,29,42,22,32,21,31)(18,23,25,37,27,38,26,28)(20,30,43,48,34,45,35,46)(24,36,33,44,39,47,40,41);(3,10)(6,9)(11,16)(14,15)(17,20,29,43,22,34,21,35)(18,24,26,40,27,39,25,33)(19,30,42,48,32,45,31,46)(23,36,28,41,38,47,37,44);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(20,34)(24,39)(30,45)(33,40)(35,43)(36,47)(41,44)(46,48)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=gi,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^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093936);