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 | 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)(21,22,26,30,27,31,39,45)(23,32,34,41,35,44,25,28)(24,29,38,43,37,42,49,51)(33,46,48,50,47,52,36,40);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,35)(22,28,31,41)(24,33,37,47)(25,39,34,26)(29,40,42,50)(30,44,45,32)(36,49,48,38)(43,52,51,46);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,29)(23,33)(25,36)(26,38)(27,37)(28,40)(30,43)(31,42)(32,46)(34,48)(35,47)(39,49)(41,50)(44,52)(45,51);(8,16)(11,20)(13,15)(17,19)(21,25,26,23,27,34,39,35)(22,28,30,32,31,41,45,44)(24,36,38,33,37,48,49,47)(29,40,43,46,42,50,51,52);(6,12)(7,14)(11,20)(13,15)(24,37)(29,42)(33,47)(36,48)(38,49)(40,50)(43,51)(46,52)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=h,d^2=f,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, 7011556);