Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1554 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)(7,13)(8,12)(9,10)(11,15)(14,20)(16,19)(17,18)(21,22,26,24)(23,29,30,27)(25,32,33,28)(31,34,36,35)(37,38)(39,44)(40,46)(41,43)(42,45)(47,51)(48,52)(49,50);(6,10)(11,13)(12,18)(19,20)(23,30)(25,33)(27,29)(28,32)(37,39,43,40)(38,44,41,46)(42,48,50,47)(45,52,49,51);(5,6)(7,13)(8,12)(9,10)(11,15)(14,20)(16,19)(17,18)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35)(37,40,43,39)(38,44,41,46)(42,48,50,47)(45,51,49,52);(1,2)(3,4)(5,7,9,15)(6,11,10,13)(8,16,17,14)(12,20,18,19)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35)(37,41,43,38)(39,46,40,44)(42,49,50,45)(47,52,48,51);(5,8,9,17)(6,12,10,18)(7,14,15,16)(11,19,13,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)(37,42,43,50)(38,45,41,49)(39,47,40,48)(44,51,46,52)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=g,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5533358);