Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1667 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)(21,22,26,30)(23,27,34,40)(24,28,36,42)(25,29,32,38)(31,37,44,48)(33,39,46,50)(35,41,45,49)(43,47,51,52);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(23,31)(24,35)(25,32)(27,37)(28,41)(29,38)(33,46)(34,44)(36,45)(39,50)(40,48)(42,49);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23,32,44,26,34,25,31)(22,27,38,48,30,40,29,37)(24,33,45,51,36,46,35,43)(28,39,49,52,42,50,41,47);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(24,36)(28,42)(33,46)(35,45)(39,50)(41,49)(43,51)(47,52);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,24,25,35,26,36,32,45)(22,28,29,41,30,42,38,49)(23,33,31,43,34,46,44,51)(27,39,37,47,40,50,48,52)Polycyclic presentation:
2^9|a^2=fi,c^2=fghi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724703);