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 | |
1538 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)(8,16,17,12)(15,18,20,19)(21,22)(23,24)(25,33)(26,27)(28,34)(29,32)(30,36)(31,35)(39,45)(40,47)(42,48)(43,50);(6,10)(7,14)(8,17)(18,19)(22,27)(23,29)(25,34)(31,36)(37,38,41,44)(39,42,45,48)(40,43,47,50)(46,49,51,52);(5,6)(7,13)(8,12)(9,10)(11,14)(15,19)(16,17)(18,20)(21,23,26,29)(22,24,27,32)(25,30,34,35)(28,31,33,36)(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)(9,14)(10,13)(12,18)(16,19)(17,20)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,35)(34,36)(37,39,41,45)(38,42,44,48)(40,46,47,51)(43,49,50,52);(5,8,9,17)(6,12,10,16)(7,15,14,20)(11,18,13,19)(21,25)(22,28)(23,30)(24,31)(26,34)(27,33)(29,35)(32,36)(37,40)(38,43)(39,46)(41,47)(42,49)(44,50)(45,51)(48,52)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fg,d^2=fh,e^2=g,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, 4376090);