Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(6,10)(7,13)(8,15)(18,20)(22,26)(23,29)(28,33)(30,35)(37,38)(39,42)(40,47)(41,44)(43,49)(45,52)(46,51)(48,50);(6,10)(7,13)(12,17)(14,19)(23,29)(27,32)(30,35)(34,36)(37,38,41,44)(39,42,46,51)(40,43,49,47)(45,50,48,52);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36)(37,38)(39,42)(40,43)(41,44)(45,50)(46,51)(47,49)(48,52);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36)(37,39,41,46)(38,42,44,51)(40,48,49,45)(43,52,47,50);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)(37,40,41,49)(38,43,44,47)(39,45,46,48)(42,50,51,52)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=fgh,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^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4776029);