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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1651 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(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)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(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)(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);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(24,36)(28,42)(33,46)(35,45)(39,50)(41,49)(43,51)(47,52);(7,15)(10,13)(14,20)(18,19)(21,24,32,45,26,36,25,35)(22,28,38,49,30,42,29,41)(23,33,44,51,34,46,31,43)(27,39,48,52,40,50,37,47)Polycyclic presentation:
2^9|a^2=fi,c^2=ghi,d^2=f,e^2=hi,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, 6722445);