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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1642 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)(7,13)(8,12)(9,10)(11,14)(15,20)(16,18)(17,19)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34)(37,38,41,44)(39,45,47,42)(40,48,49,43)(46,50,52,51);(6,10)(7,14)(8,16)(19,20)(23,29)(27,30)(31,35)(34,36)(37,38,41,44)(39,42,47,45)(40,48,49,43)(46,51,52,50);(5,7)(6,11)(8,17)(9,14)(10,13)(12,20)(15,16)(18,19)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34)(37,39)(38,42)(40,46)(41,47)(43,50)(44,45)(48,51)(49,52);(5,8,9,16)(6,12,10,18)(7,15,14,17)(11,19,13,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)(37,40,41,49)(38,43,44,48)(39,46,47,52)(42,50,45,51)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,d^2=gh,e^2=g,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, 5123694);