Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298939 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(13,14,17,20,16,19,18,15);(1,2,4,3)(5,6,7,10,8,11,12,9)(13,15,17,14,16,20,18,19);(6,9)(7,12)(10,11)(14,15)(17,18)(19,20);(1,3,4,2)(14,19)(15,20)Polycyclic presentation:
2^9|a^2=f,b^2=eg,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72226827, #72226828 |
| maximal normal subgroup of | #72226827, #72226828 |
| maximal quotient of | #72224950 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 229754);