Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2005 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)(23,32)(24,29)(25,30)(26,41)(27,31)(28,34)(33,49)(35,43)(36,52)(37,46)(38,47)(39,44)(40,48)(42,51)(45,50);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,34,27,35,25,39)(22,28,30,32,31,44,41,43)(24,36,38,50,37,33,40,51)(29,45,48,52,46,42,47,49);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(21,24)(22,29)(23,33)(25,40)(26,38)(27,37)(28,42)(30,48)(31,46)(32,49)(34,51)(35,36)(39,50)(41,47)(43,52)(44,45);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(24,37)(29,46)(33,36)(38,40)(42,45)(47,48)(49,52)(50,51);(7,13)(10,16)(14,19)(17,20)(21,25,27,26)(22,30,31,41)(23,34,35,39)(24,38,37,40)(28,43,44,32)(29,47,46,48)(33,50,36,51)(42,49,45,52)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434088);