Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(21,22,27,40,26,31,25,30)(23,32,34,28,36,42,35,43)(24,29,37,44,39,45,38,46)(33,47,50,41,49,52,48,51);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,36)(22,28,31,43)(24,33,39,49)(25,34,27,35)(29,41,45,51)(30,42,40,32)(37,48,38,50)(44,47,46,52);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24)(22,29)(23,33)(25,38)(26,39)(27,37)(28,41)(30,46)(31,45)(32,47)(34,50)(35,48)(36,49)(40,44)(42,52)(43,51);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,26,27)(22,30,31,40)(23,34,36,35)(24,37,39,38)(28,42,43,32)(29,44,45,46)(33,48,49,50)(41,47,51,52);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(22,31)(28,43)(29,45)(30,40)(32,42)(41,51)(44,46)(47,52)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,f^b=fi,d^c=di,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451459);