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 | |
1550 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)(11,16)(12,17)(18,20)(21,22)(23,29)(24,33)(25,34)(26,27)(28,32)(30,35)(31,36)(38,42)(39,45)(44,48)(47,51);(7,13)(11,16)(14,19)(18,20)(22,27)(24,34)(28,29)(30,36)(37,38,41,42)(39,46,45,43)(40,44,49,48)(47,52,51,50);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,32)(22,28,27,29)(24,30,34,36)(25,35,33,31)(37,38)(39,43)(40,48)(41,42)(44,49)(45,46)(47,52)(50,51);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24)(22,25)(23,30)(26,34)(27,33)(28,35)(29,31)(32,36)(37,39,41,45)(38,43,42,46)(40,47,49,51)(44,50,48,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25)(22,24)(23,31)(26,33)(27,34)(28,36)(29,30)(32,35)(37,40)(38,44)(39,47)(41,49)(42,48)(43,50)(45,51)(46,52)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=fh,d^2=h,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, 4517617);