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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18)(21,22,26,25)(23,29,31,27)(24,32,33,28)(30,34,36,35)(39,45)(40,47)(42,48)(43,50);(6,10)(8,16)(11,13)(14,20)(24,33)(28,32)(30,36)(34,35)(37,38,41,44)(39,42,45,48)(40,43,47,50)(46,49,51,52);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,23,26,31)(22,27,25,29)(24,30,33,36)(28,34,32,35)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(1,2)(3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,24,26,33)(22,28,25,32)(23,30,31,36)(27,34,29,35)(37,39,41,45)(38,42,44,48)(40,46,47,51)(43,49,50,52);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,25,26,22)(23,29,31,27)(24,32,33,28)(30,35,36,34)(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=h,c^2=g,d^2=fgh,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, 4376228);