Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1715 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)(21,22,26,30)(23,31,33,45)(24,28,35,42)(25,37,38,29)(27,39,41,34)(32,44,46,52)(36,49,48,43)(40,50,51,47);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,25,34,26,33,38,39)(22,27,29,31,30,41,37,45)(24,32,36,47,35,46,48,50)(28,40,43,44,42,51,49,52);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,28)(23,32)(25,36)(26,35)(27,40)(29,43)(30,42)(31,44)(33,46)(34,47)(37,49)(38,48)(39,50)(41,51)(45,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,33)(27,41)(31,45)(32,46)(34,39)(40,51)(44,52)(47,50);(6,12)(10,13)(11,20)(18,19)(24,35)(28,42)(32,46)(36,48)(40,51)(43,49)(44,52)(47,50)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437049);