Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | |
1641 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)(21,22,27,32)(23,28,36,45)(24,29,26,31)(25,38,40,30)(33,42,34,43)(35,50,51,44)(37,48,41,46)(39,52,49,47);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,36)(22,28,32,45)(24,34,26,33)(25,39,40,49)(29,43,31,42)(30,47,38,52)(35,37,51,41)(44,46,50,48);(1,2,3,4)(23,33)(24,26)(28,42)(29,31)(34,36)(35,49)(37,41)(39,51)(43,45)(44,52)(46,48)(47,50);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,24,27,26)(22,29,32,31)(23,34,36,33)(25,37,40,41)(28,43,45,42)(30,46,38,48)(35,39,51,49)(44,47,50,52);(7,15)(10,13)(14,20)(18,19)(21,25,27,40)(22,30,32,38)(23,35,36,51)(24,37,26,41)(28,44,45,50)(29,46,31,48)(33,49,34,39)(42,52,43,47)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591173);