Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1594 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)(7,13,14,11)(8,16,18,12)(15,19,17,20)(21,22)(23,29)(24,26)(25,28)(27,31)(30,35)(32,33)(34,36)(37,38,42,45)(39,40,47,43)(41,49,51,44)(46,50,52,48);(6,10)(7,14)(12,16)(15,17)(22,24)(27,29)(28,32)(34,35)(37,39,42,47)(38,43,45,40)(41,46,51,52)(44,50,49,48);(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35)(37,39)(38,40)(41,46)(42,47)(43,45)(44,48)(49,50)(51,52);(1,2)(3,4)(5,7,9,14)(6,11,10,13)(8,17,18,15)(12,20,16,19)(21,24,26,22)(23,29,31,27)(25,28,33,32)(30,34,36,35)(37,40)(38,39)(41,50)(42,43)(44,52)(45,47)(46,49)(48,51);(5,8,9,18)(6,12,10,16)(7,15,14,17)(11,19,13,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)(37,41,42,51)(38,44,45,49)(39,46,47,52)(40,48,43,50)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fh,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4773570);