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 | |
1554 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)(5,6,9,12)(7,13,14,10)(8,11,16,18)(15,19,20,17)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36)(37,38)(39,46)(40,44)(41,51)(42,52)(43,45)(47,48)(49,50);(7,14)(10,13)(15,20)(17,19)(22,26)(27,29)(28,32)(34,35)(37,39,43,40)(38,44,45,46)(41,47,52,49)(42,48,51,50);(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,31)(30,35)(34,36)(37,40,43,39)(38,44,45,46)(41,49,52,47)(42,48,51,50);(1,2)(3,4)(5,7)(6,10)(8,15)(9,14)(11,17)(12,13)(16,20)(18,19)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35)(37,41)(38,42)(39,47)(40,49)(43,52)(44,48)(45,51)(46,50);(5,8,9,16)(6,11,12,18)(7,15,14,20)(10,17,13,19)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36)(37,42)(38,41)(39,48)(40,50)(43,51)(44,47)(45,52)(46,49)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fh,d^2=h,e^2=g,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, 4509186);