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 | |
1542 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)(5,6,9,10)(7,13,15,11)(8,16,18,12)(14,19,17,20)(21,22,26,29)(23,30,27,24)(25,33,34,28)(31,32,35,36)(41,49)(44,48)(46,52)(50,51);(6,10)(11,13)(12,16)(19,20)(23,27)(24,30)(31,35)(32,36)(37,38,42,40)(39,45,47,43)(41,44,49,48)(46,51,52,50);(5,7)(6,11)(8,17)(9,15)(10,13)(12,20)(14,18)(16,19)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,18)(17,20)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36)(37,40,42,38)(39,45,47,43)(41,48,49,44)(46,51,52,50);(5,8)(6,12)(7,14)(9,18)(10,16)(11,19)(13,20)(15,17)(21,25,26,34)(22,28,29,33)(23,31,27,35)(24,32,30,36)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=h,d^2=fhi,e^2=fg,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, 4575638);