Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | #25723 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18,21,26,23,28,22,27)(19,29,32,24,31,40,33,39)(20,34,37,25,36,42,35,41)(30,38,46,43,44,47,45,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,28,39)(20,30,36,44)(21,33,22,32)(25,38,41,47)(26,29,27,40)(34,43,42,48)(35,46,37,45);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,34)(27,42)(28,41)(29,43)(31,44)(32,45)(33,46)(39,47)(40,48);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,31)(24,39)(29,40)(30,44)(32,33)(38,47)(43,48)(45,46);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,36,37)(24,40,39,29)(25,34,41,42)(30,45,44,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fgi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6470391);