Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 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)(23,31)(24,35)(25,27)(26,39)(28,34)(29,36)(30,38)(32,44)(33,47)(37,42)(40,46)(41,52)(43,45)(48,49)(50,51);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(22,27)(23,32)(29,35)(30,39)(33,45)(34,46)(42,50)(48,52);(2,4)(21,23,26,34,25,32,38,46)(22,28,30,31,27,40,39,44)(24,33,37,48,36,45,51,52)(29,41,42,43,35,49,50,47);(6,11)(10,13)(12,19)(18,20)(21,24,25,36)(22,29,27,35)(23,33,32,45)(26,37,38,51)(28,41,40,49)(30,42,39,50)(31,43,44,47)(34,48,46,52);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(24,36)(29,35)(33,45)(37,51)(41,49)(42,50)(43,47)(48,52)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593160);