Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| 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×#3257 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2009 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)(21,22,28,33)(23,34,37,30)(24,29,36,44)(25,31,40,48)(26,32,27,42)(35,49,50,45)(38,43,51,52)(39,46,41,47);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,36,28,37,27,24)(22,29,42,30,33,44,32,34)(25,35,41,51,40,50,39,38)(31,43,46,45,48,52,47,49);(7,13)(8,15)(10,16)(11,18)(21,24,27,37,28,36,26,23)(22,30,32,29,33,34,42,44)(25,38,39,50,40,51,41,35)(31,45,47,43,48,49,46,52);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,25)(22,31)(23,35)(24,38)(26,41)(27,39)(28,40)(29,43)(30,45)(32,47)(33,48)(34,49)(36,51)(37,50)(42,46)(44,52);(7,13)(10,16)(14,19)(17,20)(21,26,28,27)(22,32,33,42)(23,36,37,24)(25,39,40,41)(29,34,44,30)(31,46,48,47)(35,38,50,51)(43,45,52,49)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6434254);