Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1429 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)(17,18,22,27)(19,28,31,39)(20,32,34,42)(21,25,33,26)(23,37,40,30)(24,35,41,36)(29,44,45,38)(43,46,48,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,30,22,31,33,37)(18,23,26,28,27,40,25,39)(20,29,36,47,34,45,35,46)(24,38,32,43,41,44,42,48);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,33,35,22,34,21,36)(18,24,25,42,27,41,26,32)(19,29,37,46,31,45,30,47)(23,38,39,48,40,44,28,43);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,34)(24,41)(29,45)(32,42)(35,36)(38,44)(43,48)(46,47);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,22,33)(18,25,27,26)(19,30,31,37)(20,35,34,36)(23,39,40,28)(24,32,41,42)(29,46,45,47)(38,43,44,48)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6461441);