Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3184 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,11,14,13,19,20,18,10,16,9,15,12,7,8,17);(1,2,3,4)(5,7,10,19)(6,13,16,12)(8,17,20,18)(9,14,11,15);(8,20)(9,11)(14,15)(17,18);(5,8,13,11,10,20,12,9)(6,14,7,17,16,15,19,18)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=fh,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214114, #72214675, #72214744, … |
| maximal normal subgroup of | #72214114, #72214675, #72214744, … |
| maximal quotient of | #72216642, #72216653, #72216658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402737);