Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #39361 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C2×D42 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,13,14,20)(8,16,18,11)(9,12)(10,15)(17,19);(1,2)(3,4)(7,14)(10,19)(13,20)(15,17);(5,7,9,14)(6,10,12,19)(8,17,18,15)(11,20,16,13);(2,4)(7,14)(10,19)(13,20)(15,17)Polycyclic presentation:
2^9|a^2=g,c^2=i,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,d^c=di,f^c=fi,g^c=ghi,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72236574, #72236706, D4×#2127, … |
| maximal normal subgroup of | #72236574, #72236706, D4×#2127, … |
| maximal quotient of | #72318046, C2×#497307 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404503);