Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,27,28,26,32,25,31)(23,33,36,45,34,29,37,43)(24,38,41,48,42,30,39,44)(35,46,49,51,40,47,50,52);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(22,28)(23,34)(24,39)(25,27)(29,45)(31,32)(33,43)(35,49)(40,50)(41,42)(44,48)(46,47);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,23)(22,29)(24,40)(25,37)(26,34)(27,36)(28,43)(30,47)(31,45)(32,33)(35,42)(38,46)(39,49)(41,50)(44,52)(48,51);(8,16)(11,20)(13,15)(17,19)(21,24,25,39,26,42,27,41)(22,30,31,48,32,38,28,44)(23,35,37,50,34,40,36,49)(29,46,45,52,33,47,43,51);(6,12)(7,14)(11,20)(13,15)(21,25,26,27)(22,28,32,31)(23,36,34,37)(24,41,42,39)(29,45,33,43)(30,48,38,44)(35,50,40,49)(46,51,47,52)Polycyclic presentation:
2^9|a^2=fh,c^2=hi,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7011972);