Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| 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 | C2×#3198 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22)(23,31)(24,29)(25,37)(26,28)(27,33)(30,38)(32,44)(34,43)(35,41)(36,49)(39,47)(40,46)(42,48)(45,51)(50,52);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,25,33,26,34,38,47)(22,27,30,31,28,39,37,43)(24,32,36,46,35,45,48,52)(29,40,42,44,41,50,49,51);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(22,28)(27,39)(29,41)(30,37)(31,43)(40,50)(42,49)(44,51);(8,16)(11,20)(13,15)(17,19)(21,24,26,35)(22,29,28,41)(23,32,34,45)(25,36,38,48)(27,40,39,50)(30,42,37,49)(31,44,43,51)(33,46,47,52);(7,14)(10,18)(13,15)(17,19)(24,35)(29,41)(32,45)(36,48)(40,50)(42,49)(44,51)(46,52)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6418063);