Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1554 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)(7,13)(8,16)(9,10)(11,14)(12,17)(15,18)(19,20)(21,22,26,30)(23,24,28,27)(25,33,34,29)(31,35,36,32)(38,39)(40,47)(44,46)(48,52);(6,10)(7,14)(8,17)(18,19)(23,28)(24,27)(31,36)(32,35)(37,38,42,39)(40,45,47,43)(41,46,49,44)(48,50,52,51);(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,19)(17,20)(21,23,26,28)(22,27,30,24)(25,31,34,36)(29,35,33,32)(37,39,42,38)(40,43,47,45)(41,46,49,44)(48,50,52,51);(1,2)(3,4)(5,6)(7,13)(8,12)(9,10)(11,14)(15,19)(16,17)(18,20)(21,24,26,27)(22,28,30,23)(25,32,34,35)(29,36,33,31)(37,40,42,47)(38,43,39,45)(41,48,49,52)(44,50,46,51);(5,8)(6,12)(7,15)(9,17)(10,16)(11,18)(13,19)(14,20)(21,25,26,34)(22,29,30,33)(23,31,28,36)(24,32,27,35)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4377161);