Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1664 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)(8,15)(12,17)(13,19)(18,20)(21,22)(23,26)(24,29)(25,28)(27,31)(30,33)(32,35)(34,36)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(6,10)(11,16)(12,17)(18,20)(22,23)(24,31)(28,30)(32,36)(37,38,41,44)(39,42,47,45)(40,43,49,48)(46,50,52,51);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,23,26,22)(24,29,31,27)(25,28,33,30)(32,34,36,35)(37,38)(39,42)(40,48)(41,44)(43,49)(45,47)(46,51)(50,52);(1,2)(3,4)(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36)(37,39,41,47)(38,42,44,45)(40,46,49,52)(43,50,48,51);(5,8,9,15)(6,12,10,17)(7,13,14,19)(11,18,16,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)(37,40,41,49)(38,43,44,48)(39,46,47,52)(42,50,45,51)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=g,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4520863);