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×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22,26,31)(23,32,35,28)(24,36,39,29)(25,40,27,30)(33,43,51,49)(34,44,41,45)(37,46,50,52)(38,47,42,48);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,26,35)(22,28,31,32)(24,37,39,50)(25,34,27,41)(29,46,36,52)(30,44,40,45)(33,42,51,38)(43,47,49,48);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,26,39)(22,29,31,36)(23,33,35,51)(25,38,27,42)(28,43,32,49)(30,47,40,48)(34,50,41,37)(44,46,45,52);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,25,26,27)(22,30,31,40)(23,34,35,41)(24,38,39,42)(28,44,32,45)(29,47,36,48)(33,50,51,37)(43,46,49,52);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(23,34)(25,27)(28,45)(30,40)(32,44)(33,50)(35,41)(37,51)(38,42)(43,52)(46,49)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=f,e^2=fi,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, 6595021);