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 | |
1538 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)(5,6,10,7)(8,13,16,11)(9,12,18,14)(15,20,17,19)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34)(37,38,42,40)(39,45,47,43)(41,48,49,44)(46,50,52,51);(6,7)(11,13)(12,14)(19,20)(23,29)(27,30)(31,35)(34,36)(37,38,42,40)(39,45,47,43)(41,48,49,44)(46,50,52,51);(5,7,10,6)(8,11,16,13)(9,14,18,12)(15,19,17,20)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(1,2)(3,4)(5,8,10,16)(6,11,7,13)(9,17,18,15)(12,20,14,19)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34)(37,40,42,38)(39,43,47,45)(41,48,49,44)(46,50,52,51);(5,9,10,18)(6,12,7,14)(8,15,16,17)(11,19,13,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)(37,41,42,49)(38,44,40,48)(39,46,47,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=h,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, 5526835);