Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1721 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,25,27)(23,31,32,47)(24,36,38,29)(26,40,33,30)(28,42,41,35)(34,49,37,46)(39,45,48,51)(43,50,44,52);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(22,27)(23,32)(29,36)(30,40)(34,37)(35,42)(45,51)(50,52);(2,4)(21,23,33,42,25,32,26,35)(22,28,40,47,27,41,30,31)(24,37,48,50,38,34,39,52)(29,44,51,46,36,43,45,49);(6,11)(10,13)(12,19)(18,20)(21,24)(22,29)(23,34)(25,38)(26,39)(27,36)(28,43)(30,45)(31,46)(32,37)(33,48)(35,50)(40,51)(41,44)(42,52)(47,49);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,32)(24,38)(28,41)(29,36)(31,47)(35,42)(39,48)(45,51)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6593263);