Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2726 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)(23,27)(24,34)(25,38)(26,29)(28,37)(30,40)(31,47)(32,43)(33,50)(35,52)(36,46)(39,45)(41,48)(42,49)(44,51);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,32)(22,27,29,43)(24,35,36,31)(25,33,40,49)(28,44,45,41)(30,42,38,50)(34,52,46,47)(37,51,39,48);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,26,36)(22,28,29,45)(23,31,32,35)(25,39,40,37)(27,41,43,44)(30,46,38,34)(33,51,49,48)(42,52,50,47);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,29)(23,32)(24,36)(30,38)(33,49)(37,39)(41,44)(47,52);(6,12)(10,13)(11,20)(18,19)(22,29)(23,32)(28,45)(30,38)(31,35)(33,49)(34,46)(48,51)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436489);