Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1678 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)(21,22,27,31,28,32,26,25)(23,33,35,43,36,46,41,29)(24,30,38,44,39,45,40,37)(34,47,49,52,50,51,48,42);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,28,36)(22,29,32,43)(24,34,39,50)(25,33,31,46)(26,35,27,41)(30,42,45,52)(37,47,44,51)(38,48,40,49);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,30)(23,34)(25,37)(26,40)(27,38)(28,39)(29,42)(31,44)(32,45)(33,47)(35,49)(36,50)(41,48)(43,52)(46,51);(8,16)(11,20)(13,15)(17,19)(21,25,26,32,28,31,27,22)(23,29,41,46,36,43,35,33)(24,37,40,45,39,44,38,30)(34,42,48,51,50,52,49,47);(6,12)(7,14)(11,20)(13,15)(21,26,28,27)(22,25,32,31)(23,35,36,41)(24,38,39,40)(29,33,43,46)(30,44,45,37)(34,48,50,49)(42,51,52,47)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=h,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7011562);