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 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(8,15)(12,17)(13,19)(18,20)(21,22,25,26)(23,27,29,34)(24,31,33,28)(30,36,32,35)(38,40)(43,45)(44,48)(50,51);(6,10)(11,16)(12,17)(18,20)(22,26)(23,29)(28,31)(30,32)(37,38,42,40)(39,43,47,45)(41,44,49,48)(46,50,52,51);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,35)(31,33)(32,36)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(1,2)(3,4)(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,23,25,29)(22,27,26,34)(24,32,33,30)(28,36,31,35)(37,40,42,38)(39,45,47,43)(41,44,49,48)(46,50,52,51);(5,8)(6,12)(7,13)(9,15)(10,17)(11,18)(14,19)(16,20)(21,24,25,33)(22,28,26,31)(23,30,29,32)(27,35,34,36)(37,41,42,49)(38,44,40,48)(39,46,47,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=fg,d^2=fgh,e^2=f,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, 4775642);