Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | #25390 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10,15,14)(11,12,13,16)(17,18)(19,28)(20,33)(21,26)(22,38)(23,27)(24,32)(25,36)(29,41)(30,44)(31,40)(34,39)(35,42)(37,48)(43,46)(45,47);(2,6)(7,8)(9,11,15,13)(10,16,14,12)(17,19,22,32,23,30,21,31)(18,24,26,28,27,40,38,44)(20,34,36,45,37,29,35,46)(25,41,33,47,42,39,48,43);(1,3,5,4)(2,7,6,8)(9,12,15,16)(10,11,14,13)(17,20,21,35,23,37,22,36)(18,25,38,48,27,42,26,33)(19,29,31,45,30,34,32,46)(24,39,44,47,40,41,28,43);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,16,14,12)(19,30)(24,40)(28,44)(29,34)(31,32)(39,41)(43,47)(45,46);(9,14,15,10)(11,12,13,16)(17,21,23,22)(18,26,27,38)(19,31,30,32)(20,35,37,36)(24,28,40,44)(25,33,42,48)(29,45,34,46)(39,43,41,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fi,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476608);