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 | |
1676 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)(21,22,26,31,27,32,25,30)(23,33,36,43,37,35,41,28)(24,29,38,44,39,45,40,46)(34,47,50,52,51,48,49,42);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,37)(22,28,32,43)(24,34,39,51)(25,36,26,41)(29,42,45,52)(30,33,31,35)(38,49,40,50)(44,48,46,47);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,29)(23,34)(25,40)(26,38)(27,39)(28,42)(30,46)(31,44)(32,45)(33,47)(35,48)(36,50)(37,51)(41,49)(43,52);(8,16)(11,20)(13,15)(17,19)(21,22,25,30,27,32,26,31)(23,35,41,43,37,33,36,28)(24,29,40,46,39,45,38,44)(34,48,49,52,51,47,50,42);(6,12)(7,14)(11,20)(13,15)(21,25,27,26)(22,30,32,31)(23,36,37,41)(24,38,39,40)(28,33,43,35)(29,44,45,46)(34,49,51,50)(42,48,52,47)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,d^a=di,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, 7011914);