Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1544 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)(7,13)(8,10)(9,16)(11,15)(12,17)(14,19)(18,20)(21,22,25,26)(23,29,30,27)(24,28,32,34)(31,36,33,35)(37,38)(39,42)(40,48)(41,45)(43,47)(44,50)(46,52)(49,51);(6,8)(9,17)(11,13)(14,18)(22,26)(23,30)(24,32)(35,36)(37,39,41,47)(38,42,45,43)(40,49,50,46)(44,52,48,51);(5,7,10,15)(6,11,8,13)(9,18,17,14)(12,20,16,19)(21,23,25,30)(22,27,26,29)(24,33,32,31)(28,36,34,35)(37,39)(38,43)(40,46)(41,47)(42,45)(44,52)(48,51)(49,50);(1,2)(3,4)(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,20,18,19)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,34)(33,36)(37,38)(39,42)(40,44)(41,45)(43,47)(46,51)(48,50)(49,52);(5,9)(6,12)(7,14)(8,16)(10,17)(11,19)(13,20)(15,18)(21,24,25,32)(22,28,26,34)(23,31,30,33)(27,35,29,36)(37,40)(38,44)(39,46)(41,50)(42,51)(43,52)(45,48)(47,49)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=f,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5132866);