Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| 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×#3521 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14,17,22,19,23,18,24)(15,25,27,20,28,36,29,37)(16,30,34,21,33,40,32,39)(26,35,43,41,31,38,42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,31)(17,29)(18,27)(19,28)(21,38)(22,25)(23,37)(24,36)(26,33)(30,44)(32,42)(34,43)(35,39)(40,41);(6,9)(10,12)(13,16,19,33)(14,21,23,39)(15,26,28,31)(17,32,18,34)(20,35,37,38)(22,30,24,40)(25,41,36,44)(27,42,29,43);(6,9)(10,12)(13,17,19,18)(14,22,23,24)(15,27,28,29)(16,32,33,34)(20,36,37,25)(21,30,39,40)(26,42,31,43)(35,44,38,41);(5,7,8,11)(6,10,9,12)(14,23)(15,28)(21,39)(22,24)(26,31)(27,29)(30,40)(42,43)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398551);