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 | |
1636 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)(7,10)(8,13)(9,12)(11,16)(14,17)(15,20)(18,19)(21,22,25,28)(23,29,30,26)(24,32,33,27)(31,34,36,35)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,50)(49,52);(6,7)(9,17)(11,13)(15,18)(23,30)(24,33)(26,29)(27,32)(37,38,41,44)(39,42,47,45)(40,43,49,52)(46,50,48,51);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,20,18,19)(21,23,25,30)(22,26,28,29)(24,31,33,36)(27,34,32,35)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,50)(49,52);(1,2)(3,4)(5,8,10,16)(6,11,7,13)(9,18,17,15)(12,20,14,19)(21,22)(23,29)(24,27)(25,28)(26,30)(31,35)(32,33)(34,36)(37,39)(38,42)(40,48)(41,47)(43,51)(44,45)(46,49)(50,52);(5,9)(6,12)(7,14)(8,15)(10,17)(11,19)(13,20)(16,18)(21,24,25,33)(22,27,28,32)(23,31,30,36)(26,34,29,35)(37,40,41,49)(38,43,44,52)(39,46,47,48)(42,50,45,51)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,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, 5525180);