Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(10,13)(11,15)(17,18,22,24)(19,29,30,41)(20,26,33,38)(21,28,23,27)(25,35,37,32)(31,43,44,48)(34,42,36,40)(39,45,47,46);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(18,24)(19,30)(20,33)(27,28)(32,35)(34,36)(39,47)(43,48);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,35,22,30,23,32)(18,25,27,41,24,37,28,29)(20,31,36,45,33,44,34,46)(26,39,42,48,38,47,40,43);(3,8)(6,7)(10,13)(14,16)(17,20)(18,26)(19,31)(21,36)(22,33)(23,34)(24,38)(25,39)(27,42)(28,40)(29,43)(30,44)(32,46)(35,45)(37,47)(41,48);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,24,28)(19,32,30,35)(20,34,33,36)(25,29,37,41)(26,40,38,42)(31,45,44,46)(39,48,47,43)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=fg,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6956975);