Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #38948 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2044 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,26,30)(23,31,34,45)(24,28,36,42)(25,38,32,29)(27,39,41,35)(33,44,47,52)(37,49,46,43)(40,50,51,48);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,32,39,26,34,25,35)(22,27,38,45,30,41,29,31)(24,33,46,50,36,47,37,48)(28,40,49,52,42,51,43,44);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,28)(23,33)(25,37)(26,36)(27,40)(29,43)(30,42)(31,44)(32,46)(34,47)(35,48)(38,49)(39,50)(41,51)(45,52);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(23,34)(24,36)(27,41)(28,42)(31,45)(35,39)(37,46)(43,49);(8,15)(11,18)(14,19)(17,20)(24,36)(28,42)(33,47)(37,46)(40,51)(43,49)(44,52)(48,50)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476794);