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 | |
1672 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)(8,15)(11,16)(14,19)(21,22,25,26)(23,27,29,34)(24,31,33,28)(30,36,32,35)(40,47)(43,50)(45,51)(48,52);(6,10)(7,13)(12,17)(14,19)(22,26)(23,29)(28,31)(30,32)(37,38,41,44)(39,42,46,49)(40,43,47,50)(45,48,51,52);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,35)(31,33)(32,36)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23)(22,27)(24,32)(25,29)(26,34)(28,36)(30,33)(31,35)(37,39,41,46)(38,42,44,49)(40,45,47,51)(43,48,50,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24)(22,28)(23,30)(25,33)(26,31)(27,35)(29,32)(34,36)(37,40)(38,43)(39,45)(41,47)(42,48)(44,50)(46,51)(49,52)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=fg,d^2=fh,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, 4775930);