Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1732 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)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22,26,30)(23,31,35,45)(24,28,36,42)(25,38,32,29)(27,39,41,34)(33,44,48,52)(37,49,46,43)(40,50,51,47);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,32,39,26,35,25,34)(22,27,38,45,30,41,29,31)(24,33,46,50,36,48,37,47)(28,40,49,52,42,51,43,44);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,26,36)(22,28,30,42)(23,33,35,48)(25,37,32,46)(27,40,41,51)(29,43,38,49)(31,44,45,52)(34,47,39,50);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,34)(25,32)(27,31)(29,38)(33,47)(35,39)(37,46)(40,44)(41,45)(43,49)(48,50)(51,52);(6,12)(10,13)(11,20)(18,19)(23,35)(24,36)(27,41)(28,42)(31,45)(34,39)(37,46)(43,49)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=hi,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7032875);