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 | |
1586 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)(6,10)(7,13)(12,17)(15,19)(21,22)(23,30)(24,25)(26,27)(28,33)(29,34)(31,36)(32,35)(37,38,41,45)(39,43,47,42)(40,44,49,48)(46,51,52,50);(8,16)(12,17)(15,19)(18,20)(22,27)(25,34)(28,30)(32,36)(37,39,41,47)(38,42,45,43)(40,46,49,52)(44,50,48,51);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,23,26,33)(22,28,27,30)(24,31,29,35)(25,32,34,36)(37,39)(38,43)(40,46)(41,47)(42,45)(44,51)(48,50)(49,52);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,24)(22,25)(23,31)(26,29)(27,34)(28,32)(30,36)(33,35)(37,38)(39,42)(40,48)(41,45)(43,47)(44,49)(46,51)(50,52);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,25)(22,29)(23,32)(24,27)(26,34)(28,35)(30,31)(33,36)(37,40,41,49)(38,44,45,48)(39,46,47,52)(42,50,43,51)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=fgh,e^2=fg,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, 5525115);