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)(23,29)(24,26)(25,28)(27,30)(31,36)(32,34)(33,35)(37,38)(39,46)(40,44)(41,51)(42,52)(43,45)(47,48)(49,50);(6,10)(11,13)(12,16)(19,20)(22,24)(23,30)(28,32)(31,33)(37,39,43,40)(38,44,45,46)(41,47,52,49)(42,48,51,50);(5,7)(6,11)(8,17)(9,15)(10,13)(12,20)(14,18)(16,19)(21,23)(22,27)(24,29)(25,33)(26,30)(28,36)(31,34)(32,35)(37,40,43,39)(38,44,45,46)(41,49,52,47)(42,48,51,50);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,18)(17,20)(21,24,26,22)(23,29,30,27)(25,32,34,28)(31,36,33,35)(37,41)(38,42)(39,47)(40,49)(43,52)(44,48)(45,51)(46,50);(5,8)(6,12)(7,14)(9,18)(10,16)(11,19)(13,20)(15,17)(21,25,26,34)(22,28,24,32)(23,31,30,33)(27,35,29,36)(37,42)(38,41)(39,48)(40,50)(43,51)(44,47)(45,52)(46,49)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fg,d^2=fhi,e^2=h,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, 4579082);