Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1723 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,22,26,31)(23,32,35,45)(24,28,36,43)(25,29,39,30)(27,38,41,34)(33,46,47,51)(37,44,49,42)(40,50,52,48);(2,4)(21,23,26,35)(22,27,31,41)(24,33,36,47)(25,38,39,34)(28,40,43,52)(29,32,30,45)(37,50,49,48)(42,51,44,46);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24)(22,28)(23,33)(25,37)(26,36)(27,40)(29,44)(30,42)(31,43)(32,46)(34,48)(35,47)(38,50)(39,49)(41,52)(45,51);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(22,29)(24,36)(25,39)(27,32)(28,42)(30,31)(33,47)(34,38)(40,51)(41,45)(43,44)(46,52)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584700);