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)(8,15)(12,17)(13,19)(18,20)(21,22,25,26)(23,29,30,27)(24,28,33,32)(31,35,36,34)(37,38)(39,42)(40,43)(41,44)(45,49)(46,50)(47,48)(51,52);(6,10)(11,16)(12,17)(18,20)(22,26)(23,30)(28,32)(31,36)(37,38,41,44)(39,42,46,50)(40,43,48,47)(45,49,52,51);(5,6,9,10)(7,11,14,16)(8,12,15,17)(13,18,19,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,35)(34,36)(37,38)(39,42)(40,47)(41,44)(43,48)(45,51)(46,50)(49,52);(1,2)(3,4)(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,23,25,30)(22,27,26,29)(24,31,33,36)(28,34,32,35)(37,39,41,46)(38,42,44,50)(40,45,48,52)(43,49,47,51);(5,8)(6,12)(7,13)(9,15)(10,17)(11,18)(14,19)(16,20)(21,24,25,33)(22,28,26,32)(23,31,30,36)(27,34,29,35)(37,40)(38,43)(39,45)(41,48)(42,49)(44,47)(46,52)(50,51)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fh,d^2=h,e^2=g,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, 4517122);