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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1609 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)(5,6,10,8)(7,13,15,11)(9,12,17,16)(14,19,20,18)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36)(37,38,41,44)(39,45,47,42)(40,43,50,49)(46,52,48,51);(9,17)(12,16)(14,20)(18,19)(22,25)(27,29)(28,32)(34,35)(37,39,41,47)(38,42,44,45)(40,48,50,46)(43,52,49,51);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,23)(22,27)(24,30)(25,29)(26,31)(28,34)(32,35)(33,36)(37,39)(38,42)(40,46)(41,47)(43,51)(44,45)(48,50)(49,52);(1,2)(3,4)(5,8,10,6)(7,11,15,13)(9,16,17,12)(14,18,20,19)(21,24,26,33)(22,28,25,32)(23,30,31,36)(27,34,29,35)(37,38)(39,42)(40,49)(41,44)(43,50)(45,47)(46,52)(48,51);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)(21,25,26,22)(23,29,31,27)(24,28,33,32)(30,34,36,35)(37,40,41,50)(38,43,44,49)(39,46,47,48)(42,51,45,52)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fg,d^2=gh,e^2=g,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, 5525128);