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 | |
1556 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)(7,11)(8,10)(9,12)(13,19)(14,15)(16,18)(17,20)(21,22,26,29)(23,24,31,27)(25,28,33,35)(30,32,36,34)(39,45)(40,47)(42,48)(43,50);(6,8)(11,15)(12,16)(19,20)(25,33)(28,35)(30,36)(32,34)(37,38,41,44)(39,42,45,48)(40,43,47,50)(46,49,51,52);(5,7,10,14)(6,11,8,15)(9,17,18,13)(12,20,16,19)(21,23,26,31)(22,27,29,24)(25,30,33,36)(28,34,35,32)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(1,2,3,4)(5,8,10,6)(7,11,14,15)(9,16,18,12)(13,19,17,20)(21,24,26,27)(22,23,29,31)(25,32,33,34)(28,30,35,36)(37,39)(38,42)(40,46)(41,45)(43,49)(44,48)(47,51)(50,52);(5,9,10,18)(6,12,8,16)(7,13,14,17)(11,19,15,20)(21,25,26,33)(22,28,29,35)(23,30,31,36)(24,32,27,34)(37,40,41,47)(38,43,44,50)(39,46,45,51)(42,49,48,52)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=hi,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, 4992071);