Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#493 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,10,8,11,7,12)(13,14,17,19,18,20,15,16);(5,6,7,12,8,11,9,10)(14,16)(15,17)(19,20);(1,2,3,4)(5,7,8,9)(6,10,11,12)(13,15,18,17)(14,16,20,19);(5,7,8,9)(6,10,11,12)(13,16,17,14,18,19,15,20)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de,e^b=eh,g^b=gh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72343120, #72343147, #72343162, … |
| maximal normal subgroup of | #72343120, #72343147, #72343162, … |
| maximal quotient of | #72343443, #72343473 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138638);