Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #39131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(21,22)(23,31)(24,35)(25,38)(26,29)(27,34)(28,36)(30,41)(32,42)(33,45)(37,46)(39,43)(40,44)(47,49)(48,51)(50,52);(2,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,33)(22,27,29,43)(24,32,37,48)(25,39,41,34)(28,42,44,51)(30,45,38,31)(35,47,46,52)(36,50,40,49);(6,10)(11,17)(12,18)(19,20)(21,24,26,37)(22,28,29,44)(23,32,33,48)(25,40,41,36)(27,42,43,51)(30,46,38,35)(31,47,45,52)(34,50,39,49);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(23,33)(27,43)(31,45)(32,48)(34,39)(42,51)(47,52)(49,50);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(22,29)(27,43)(28,44)(30,38)(31,45)(35,46)(42,51)(47,52)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451290);