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 | #45243 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1488 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,30)(21,26)(22,36)(23,27)(25,33)(28,39)(29,37)(31,42)(32,43)(34,40)(35,46)(38,44)(41,47)(45,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,36)(20,28,32,33)(21,29,34,35)(25,30,39,43)(26,37,40,46)(31,41,45,44)(38,42,47,48);(9,11)(10,13)(12,16)(14,15)(17,20,19,28,23,32,22,33)(18,25,36,43,27,39,24,30)(21,31,29,41,34,45,35,44)(26,38,46,48,40,47,37,42);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21)(18,26)(19,29)(20,31)(22,35)(23,34)(24,37)(25,38)(27,40)(28,41)(30,42)(32,45)(33,44)(36,46)(39,47)(43,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(20,32)(21,34)(25,39)(26,40)(28,33)(29,35)(30,43)(37,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=f,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=eh,d^c=dh,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, 6936887);