Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1540 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)(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22)(23,27)(24,26)(25,28)(29,35)(30,31)(32,34)(33,36)(37,38)(39,43)(40,42)(41,48)(44,47)(45,50)(46,52)(49,51);(8,16)(11,18)(14,20)(17,19)(22,24)(27,31)(28,32)(35,36)(37,39,42,47)(38,43,40,44)(41,49,50,46)(45,52,48,51);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,30)(22,27,24,31)(25,33,34,29)(28,36,32,35)(37,39)(38,44)(40,43)(41,46)(42,47)(45,52)(48,51)(49,50);(1,2,3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,24,26,22)(23,27,30,31)(25,32,34,28)(29,35,33,36)(37,40,42,38)(39,44,47,43)(41,48,50,45)(46,52,49,51);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,25,26,34)(22,28,24,32)(23,29,30,33)(27,35,31,36)(37,41)(38,45)(39,46)(40,48)(42,50)(43,51)(44,52)(47,49)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=f,d^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4991075);