Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994171 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,17)(15,19)(18,20);(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,18)(17,19);(2,4)(5,7,8,12)(6,9,10,11);(7,12)(9,11)(14,17)(15,19)(18,20);(6,10)(9,11)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dh,f^d=fi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72281878, #72282003, #72282751, … |
| maximal normal subgroup of | #72281878, #72282003, #72282751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970027);