Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| 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×#1619 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1086 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1086 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,9)(7,12)(10,11)(13,14,15,17)(16,18,20,19);(1,3,4,2)(5,6,8,11)(7,10,12,9)(14,17)(18,19);(14,18)(16,20)(17,19);(6,10)(7,12)(9,11)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dei,e^b=eh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72214525, #72279339, #72279390, … |
| maximal normal subgroup of | #72214525, #72279339, #72279390, … |
| maximal quotient of | #72214217, #72214250, #72216495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134274);