Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(6,10)(7,13)(12,17)(14,19)(21,22,26,27)(23,30,33,28)(24,34,29,25)(31,35,36,32)(38,42)(43,45)(44,48)(50,51);(7,13)(11,16)(14,19)(18,20)(22,27)(24,29)(28,30)(31,36)(37,38,41,42)(39,45,47,43)(40,44,49,48)(46,51,52,50);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,33)(22,28,27,30)(24,31,29,36)(25,35,34,32)(37,38)(39,43)(40,48)(41,42)(44,49)(45,47)(46,51)(50,52);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,29)(22,25,27,34)(23,31,33,36)(28,35,30,32)(37,39,41,47)(38,43,42,45)(40,46,49,52)(44,50,48,51);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25)(22,29)(23,32)(24,27)(26,34)(28,31)(30,36)(33,35)(37,40)(38,44)(39,46)(41,49)(42,48)(43,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4515545);