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)(5,6,9,10)(7,11,15,13)(8,12,17,18)(14,19,16,20)(21,22,26,29)(23,24,31,27)(25,28,33,35)(30,32,36,34)(37,38)(39,43)(40,42)(41,48)(44,47)(45,50)(46,52)(49,51);(6,10)(11,13)(12,18)(19,20)(25,33)(28,35)(30,36)(32,34)(37,39,42,47)(38,43,40,44)(41,49,50,46)(45,52,48,51);(5,7)(6,11)(8,16)(9,15)(10,13)(12,20)(14,17)(18,19)(21,23,26,31)(22,27,29,24)(25,30,33,36)(28,34,35,32)(37,39)(38,44)(40,43)(41,46)(42,47)(45,52)(48,51)(49,50);(1,2,3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,20)(16,19)(17,18)(21,24,26,27)(22,23,29,31)(25,32,33,34)(28,30,35,36)(37,40,42,38)(39,44,47,43)(41,48,50,45)(46,52,49,51);(5,8,9,17)(6,12,10,18)(7,14,15,16)(11,19,13,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)(37,41)(38,45)(39,46)(40,48)(42,50)(43,51)(44,52)(47,49)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=gh,d^2=fi,e^2=f,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, 4989688);