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×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1747 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)(23,32)(24,36)(25,40)(26,31)(27,30)(28,35)(29,39)(33,43)(34,44)(37,46)(38,47)(41,45)(42,48)(49,51)(50,52);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,28)(24,37)(25,34)(26,35)(27,41)(29,46)(30,44)(31,32)(33,38)(36,52)(39,50)(40,45)(42,51)(43,48)(47,49);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24)(22,29)(23,33)(25,38)(26,39)(27,42)(28,43)(30,47)(31,36)(32,49)(34,50)(35,51)(37,41)(40,48)(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=fi,c^2=fg,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, 6595025);