Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1987 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,31)(24,35)(25,38)(26,28)(27,34)(29,36)(30,39)(32,44)(33,47)(37,42)(40,52)(41,46)(43,45)(48,49)(50,51);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,25,34,26,32,39,46)(22,27,30,31,28,41,38,44)(24,33,37,48,36,45,51,52)(29,40,42,43,35,49,50,47);(1,2,3,4)(7,13)(10,17)(14,19)(18,20)(22,28)(23,32)(29,35)(30,38)(33,45)(34,46)(42,50)(48,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,24,26,36)(22,29,28,35)(23,33,32,45)(25,37,39,51)(27,40,41,49)(30,42,38,50)(31,43,44,47)(34,48,46,52);(24,36)(29,35)(33,45)(37,51)(40,49)(42,50)(43,47)(48,52)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,d^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343596);