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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1611 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(6,10)(7,13)(12,17)(15,19)(21,22)(23,27)(24,26)(25,28)(29,35)(30,31)(32,34)(33,36)(39,45)(40,47)(42,48)(43,50);(8,16)(12,17)(15,19)(18,20)(22,24)(27,31)(28,32)(35,36)(37,38,41,44)(39,42,45,48)(40,43,47,50)(46,49,51,52);(5,6)(7,14)(8,12)(9,10)(11,13)(15,20)(16,17)(18,19)(21,23,26,30)(22,27,24,31)(25,33,34,29)(28,36,32,35)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(1,2,3,4)(5,7)(6,11)(8,15)(9,13)(10,14)(12,18)(16,19)(17,20)(21,24,26,22)(23,27,30,31)(25,32,34,28)(29,35,33,36)(37,39)(38,42)(40,46)(41,45)(43,49)(44,48)(47,51)(50,52);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,25,26,34)(22,28,24,32)(23,29,30,33)(27,35,31,36)(37,40,41,47)(38,43,44,50)(39,46,45,51)(42,49,48,52)Polycyclic presentation:
2^9|b^2=h,c^2=f,d^2=ghi,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, 4992095);