Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #2397 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,28)(24,34)(25,30)(26,40)(27,31)(29,37)(32,43)(33,41)(35,46)(36,47)(38,44)(39,50)(42,48)(45,51)(49,52);(2,3)(7,13)(8,16)(10,17)(11,15)(21,23,27,26)(22,28,31,40)(24,32,36,37)(25,33,38,39)(29,34,43,47)(30,41,44,50)(35,45,49,48)(42,46,51,52);(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24,26,37,27,36,23,32)(22,29,28,34,31,43,40,47)(25,35,39,48,38,49,33,45)(30,42,41,46,44,51,50,52);(1,3,4,2)(7,13)(10,17)(14,19)(18,20)(21,25,27,38)(22,30,31,44)(23,33,26,39)(24,35,36,49)(28,41,40,50)(29,42,43,51)(32,45,37,48)(34,46,47,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,36)(25,38)(29,43)(30,44)(32,37)(33,39)(34,47)(41,50)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=gh,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104085);