Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(7,13)(8,15)(11,16)(12,17)(21,22,25,26)(23,29,30,27)(24,28,32,34)(31,36,33,35)(37,38)(39,42)(40,48)(41,45)(43,47)(44,50)(46,52)(49,51);(6,10)(11,16)(12,17)(18,20)(22,26)(23,30)(24,32)(35,36)(37,39,41,47)(38,42,45,43)(40,49,50,46)(44,52,48,51);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,25,30)(22,27,26,29)(24,33,32,31)(28,36,34,35)(37,39)(38,43)(40,46)(41,47)(42,45)(44,52)(48,51)(49,50);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,34)(33,36)(37,38)(39,42)(40,44)(41,45)(43,47)(46,51)(48,50)(49,52);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,24)(22,28)(23,31)(25,32)(26,34)(27,35)(29,36)(30,33)(37,40,41,50)(38,44,45,48)(39,46,47,49)(42,51,43,52)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5123202);