Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129412 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #35513 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1824 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1824 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,11,8,13)(9,14)(10,15)(12,16,19,17)(18,20);(1,2)(3,4)(6,11)(7,8)(12,20)(13,15)(14,19)(16,17);(2,4)(5,7,10,8)(6,12,15,19)(9,17,18,16)(11,14,13,20)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,g^b=ghi,e^c=ei,g^c=gi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219502, #72231758, #72238623, … |
| maximal normal subgroup of | #72219502, #72231758, #72238623, … |
| maximal quotient of | #72219498, #72231768, #72238662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54772);