Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22)(23,32)(24,36)(25,31)(26,28)(27,40)(29,35)(30,38)(33,44)(34,49)(37,46)(39,45)(41,42)(43,50)(47,48)(51,52);(1,2)(3,4)(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(22,28)(23,33)(30,36)(31,40)(34,48)(35,41)(45,46)(50,51);(2,4)(21,23,26,33)(22,29,28,42)(24,34,38,48)(25,35,27,41)(30,43,36,52)(31,44,40,32)(37,50,39,51)(45,47,46,49);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24)(22,30)(23,34)(25,39)(26,38)(27,37)(28,36)(29,43)(31,46)(32,47)(33,48)(35,51)(40,45)(41,50)(42,52)(44,49);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,26,27)(22,31,28,40)(23,35,33,41)(24,37,38,39)(29,44,42,32)(30,45,36,46)(34,50,48,51)(43,47,52,49)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584846);