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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(8,15)(12,17)(13,19)(18,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36)(38,39)(41,49)(43,45)(47,52);(6,10)(11,16)(12,17)(18,20)(22,26)(27,29)(28,32)(34,35)(37,38,42,39)(40,45,48,43)(41,44,49,46)(47,51,52,50);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,31)(30,35)(34,36)(37,39,42,38)(40,45,48,43)(41,44,49,46)(47,50,52,51);(1,2)(3,4)(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36)(37,40)(38,43)(39,45)(41,47)(42,48)(44,50)(46,51)(49,52);(5,8,9,15)(6,12,10,17)(7,13,14,19)(11,18,16,20)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)(37,41,42,49)(38,44,39,46)(40,47,48,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=g,d^2=fgh,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, 4520901);