Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#1712 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
954 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 954 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,8,10,9,11,12,7)(14,17)(15,19)(18,20);(1,3,4,2)(6,7)(8,12)(10,11)(14,18)(17,20);(13,14,15,17,16,18,19,20);(5,7,8,6,9,10,12,11)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=ci,d^b=de,e^b=eh,g^b=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214397, #72214525, #72214723, … |
| maximal normal subgroup of | #72214397, #72214525, #72214723, … |
| maximal quotient of | #72214332, #72216429, #72227103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138827);