Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499771 |
| 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 | #34909 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(21,22)(23,31)(24,30)(25,28)(26,29)(27,32)(33,36)(34,35);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,34)(22,27,29,36)(24,33,25,32)(28,35,30,31);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(22,28)(24,25)(27,35)(29,30)(31,36)(32,33);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,29)(27,36)(28,30)(31,35);(6,12)(10,13)(11,20)(18,19)(22,29)(27,36)(28,30)(31,35)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,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, 6436319);