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 | |
1636 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)(5,6,9,12)(7,13,14,10)(8,11,16,18)(15,19,20,17)(21,22)(23,27)(24,32)(25,26)(28,34)(29,31)(30,36)(33,35)(37,38,41,45)(39,43,47,42)(40,44,49,52)(46,51,48,50);(7,14)(10,13)(15,20)(17,19)(22,25)(23,29)(28,32)(30,33)(37,39,41,47)(38,42,45,43)(40,48,49,46)(44,51,52,50);(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,29)(22,27,25,31)(24,33,34,30)(28,36,32,35)(37,38)(39,42)(40,44)(41,45)(43,47)(46,50)(48,51)(49,52);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,17)(12,13)(16,20)(18,19)(21,24)(22,28)(23,30)(25,32)(26,34)(27,35)(29,33)(31,36)(37,39)(38,43)(40,46)(41,47)(42,45)(44,51)(48,49)(50,52);(5,8,9,16)(6,11,12,18)(7,15,14,20)(10,17,13,19)(21,25,26,22)(23,31,29,27)(24,32,34,28)(30,36,33,35)(37,40)(38,44)(39,46)(41,49)(42,50)(43,51)(45,52)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=h,d^2=fgi,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, 4408118);