Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C22×#547 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,4,3)(5,6,9,8)(7,10,11,12)(13,14,17,20)(15,18,16,19);(13,14)(15,18)(16,19)(17,20);(6,8)(10,12);(7,11)(10,12)(13,15,17,16)(14,18,20,19);(5,7)(6,10)(8,12)(9,11)(13,15,17,16)(14,18,20,19);(1,3,4,2)(5,8,9,6)(7,12,11,10)(13,16,17,15)(14,19,20,18)Polycyclic presentation:
2^9|a^2=i,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,f^c=fh,e^d=eh170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533062);