Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| 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×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2940 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)(7,11)(10,12)(13,14,17,16,18,19,15,20);(6,9)(7,11)(13,14)(15,16)(17,20)(18,19);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,16,19,20);(6,9)(10,12)(13,16,15,14,18,20,17,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251262, #72251275, #72256831, … |
| maximal normal subgroup of | #72251262, #72251275, #72256831, … |
| maximal quotient of | #72264746, #72264948 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7052911);