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)(7,13)(8,15)(11,16)(12,17)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36)(37,38,41,42)(39,45,47,43)(40,44,48,49)(46,51,52,50);(6,10)(11,16)(12,17)(18,20)(22,26)(24,33)(27,29)(30,36)(38,42)(40,48)(43,45)(46,52);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36)(37,39,41,47)(38,43,42,45)(40,46,48,52)(44,50,49,51);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35)(37,38)(39,45)(40,44)(41,42)(43,47)(46,51)(48,49)(50,52);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)(37,40)(38,44)(39,46)(41,48)(42,49)(43,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,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, 4375388);