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)(5,6,9,10)(7,13,15,11)(8,12,16,18)(14,20,17,19)(21,22,26,24)(23,29,30,27)(25,32,33,28)(31,34,36,35)(37,38,42,40)(39,43,47,45)(41,48,49,44)(46,51,52,50);(6,10)(8,16)(11,13)(14,17)(23,30)(25,33)(27,29)(28,32)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,18)(17,20)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35)(37,39)(38,43)(40,45)(41,46)(42,47)(44,50)(48,51)(49,52);(1,2)(3,4)(5,7,9,15)(6,11,10,13)(8,17,16,14)(12,20,18,19)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35)(37,40,42,38)(39,45,47,43)(41,48,49,44)(46,51,52,50);(5,8,9,16)(6,12,10,18)(7,14,15,17)(11,19,13,20)(21,25,26,33)(22,28,24,32)(23,31,30,36)(27,34,29,35)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fg,d^2=h,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, 5524903);