Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10,14,11)(12,15,13,16)(17,18,22,27)(19,28,23,24)(20,31,33,38)(21,26,37,44)(25,29,41,34)(30,42,35,39)(32,47,36,48)(40,45,43,46);(1,2)(3,7)(4,8)(5,6)(10,11)(12,13)(17,19,22,23)(18,24,27,28)(20,29,33,34)(21,35,37,30)(25,38,41,31)(26,42,44,39)(32,46,36,45)(40,47,43,48);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,15,13,16)(17,20,19,29,22,33,23,34)(18,25,24,38,27,41,28,31)(21,36,30,46,37,32,35,45)(26,43,39,47,44,40,42,48);(1,3,5,4)(2,8,6,7)(9,12,14,13)(10,15,11,16)(17,21,22,37)(18,26,27,44)(19,30,23,35)(20,32,33,36)(24,39,28,42)(25,40,41,43)(29,45,34,46)(31,47,38,48);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,15,11,16)(21,37)(26,44)(30,35)(32,36)(39,42)(40,43)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=fg,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6959344);