Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1556 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)(8,16,18,12)(14,20,17,19)(21,22)(23,29)(24,31)(25,33)(26,27)(28,34)(30,32)(35,36)(37,38,41,44)(39,45,47,42)(40,48,49,43)(46,50,52,51);(6,10)(11,13)(12,16)(19,20)(22,27)(24,29)(25,34)(30,35)(37,38,41,44)(39,42,47,45)(40,48,49,43)(46,51,52,50);(5,7,9,15)(6,11,10,13)(8,17,18,14)(12,20,16,19)(21,23)(22,24)(25,35)(26,31)(27,29)(28,36)(30,34)(32,33)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(1,2,3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,20)(16,18)(17,19)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,35)(34,36)(37,39,41,47)(38,42,44,45)(40,46,49,52)(43,50,48,51);(5,8,9,18)(6,12,10,16)(7,14,15,17)(11,19,13,20)(21,25)(22,28)(23,30)(24,32)(26,34)(27,33)(29,36)(31,35)(37,40,41,49)(38,43,44,48)(39,46,47,52)(42,50,45,51)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fgh,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5188120);