Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×#447 |
| 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 | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,17,20)(15,21,23,28)(16,24,26,19)(18,25,27,22);(6,10)(8,11)(13,15,17,23)(14,18,20,27)(16,25,26,22)(19,28,24,21);(2,4)(5,6)(7,8)(9,10)(11,12);(5,7,9,12)(6,11,10,8);(5,8,9,11)(6,7,10,12)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=eh95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524369);