Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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 | #36015 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1682 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,24,20)(19,25,31,33)(21,26,37,34)(22,27,23,28)(29,38,41,32)(30,40,46,44)(35,39,42,36)(43,45,48,47);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,24,31)(18,25,20,33)(21,30,37,46)(22,32,23,38)(26,40,34,44)(27,29,28,41)(35,43,42,48)(36,47,39,45);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,20,24,18)(19,29,31,41)(21,35,37,42)(22,27,23,28)(25,38,33,32)(26,39,34,36)(30,44,46,40)(43,45,48,47);(9,12,13,11)(10,16,15,14)(17,21,24,37)(18,26,20,34)(19,30,31,46)(22,36,23,39)(25,40,33,44)(27,35,28,42)(29,43,41,48)(32,47,38,45);(1,4,5,3)(2,6,8,7)(17,22)(18,27)(19,31)(20,28)(23,24)(25,33)(30,45)(35,42)(36,39)(40,48)(43,44)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=fi,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6497890);